6. Exponential and Logarithmic Functions

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log5 (7 × 3)

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log7 (7x)

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log7 (7/x)

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log(x/100)

Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 10^12

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln(e^2/5)

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb x^3

Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 0.0022

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln 5√x (fifth root of)

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb (x^2 y)

Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 387 + log 23

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log6 (36/(√(x+1))

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb ((x^2 y)/z^2)

For each substance, find the pH from the given hydronium ion concentration to the nearest tenth. See Example 2(a). limes, 1.6*10^-2

Use a calculator to find an approximation to four decimal places for each logarithm. ln 144,000

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log ∛(x/y)

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb ((√x y^3)/z^3)

Find the [H_3O^+] for each substance with the given pH. Write answers in scientific notation to the nearest tenth. See Example 2(b). beer, 4.8

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln[(x^3(√(x^2 + 1))/(x + 1)^4]

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log [(10x^2∛(1 - x))/(7(x + 1)^2)]

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Determine whether the wetland is a rich fen, a poor fen, or a bog. See Example 3. 2.49*10^-7

Solve each problem. Use a calculator to find an approximation for each logarithm. log 398.4

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. ln x + ln 7

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log2 (96) - log2 (3)

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 1/e^2

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log(2x + 5) - log x

In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log4 (√x/64)

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. (1/2)ln x + ln y

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 2 logb x + 3 logb y

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 0.00013

In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. log 3 - 3 log x

In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. 1/2 ln x - ln y

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 3 ln x - (1/3) ln y

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 4 ln (x + 6) - 3 ln x

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 27 + ln 943

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. (1/2)(log x + log y)

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. (1/2)(log5 x + log5 y) - 2 log5 (x + 1)

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log x + log(x^2 - 1) - log 7 - log(x + 1)

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log5 13

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log0.1 17

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. logπ 63

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_2 5

In Exercises 79–82, use a graphing utility and the change-of-base property to graph each function. y = log3 x

In Exercises 79–82, use a graphing utility and the change-of-base property to graph each function. y = log2 (x + 2)

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_π e

In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C. logb 8

In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C. logb √(2/27)

Solve: log₂ (x+9) — log₂ x = 1.

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(8x^3) = 3 ln (2x)

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. x log 10^x = x^2

Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln √(a^3/b^5)

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given g(x) = e^x, find g(ln 1/e)

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(5x) + ln 1 = ln(5x)

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln x + ln(2x) = ln(3x)

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = 3^x, find ƒ(log_3 (2 ln 3))

Retaining the Concepts. Expand: log7 (5√x/49y^10) fifth root of x

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log_2 x, find ƒ(2^(2 log_2 2))

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. [log(x + 2)/log(x - 1)] = log(x + 2) - log(x - 1)

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log6 [(x - 1)/(x^2 + 4)] = log6 (x - 1) - log6 (x^2 + 4)

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log3 (7) = 1/[log7 (3)]

Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log￬3 [9 (x+2) ]

Use properties of logarithms to rewrite each function, and describe how the graph of the given function compares to the graph of g(x) = ln x. ƒ(x) = ln(e^2x)

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. log7 49 / log7 7 = log7 49 - log7 7

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. logb (xy)^5 = (logb x + logb y)^5

If log 3 = A and log 7 = B, find log7 (9) in terms of A and B.

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log(1000x)

Answer each of the following. Between what two consecutive integers must log_2 12 lie?

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log4 (64/y)

Find each value. If applicable, give an approximation to four decimal places. See Example 1. . log 63

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log N^(-6)

Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 518/342

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log4 (√x/64)

For each substance, find the pH from the given hydronium ion concentration to the nearest tenth. See Example 2(a). grapefruit, 6.3*10^-4

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log √(100x)

Find the [H_3O^+] for each substance with the given pH. Write answers in scientific notation to the nearest tenth. See Example 2(b). soda pop, 2.7

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log5 ∛((x^2 y)/24)

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Determine whether the wetland is a rich fen, a poor fen, or a bog. See Example 3. 2.49*10^-2

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log 5 + log 2

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln e^1.6

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log x + 3 log y

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 28

In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln ∛(x/e)

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 5 ln x - 2 ln y

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 98/13

In Exercises 58–59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log4 0.863

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 3 ln x + 5 ln y - 6 ln z

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 84 - ln 17

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. (1/3) [2 ln(x + 5) - ln x - ln (x^2 - 4)]

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log14 87.5

Expand: log8((4√x)/(64y3))

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. . log_1/2 3

In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C. logb (3/2)

Solve: log₂ (x+9) — log₂ x = 1.

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_√19 5

Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln (b^4√a)

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log4 (2x^3) = 3 log4 (2x)

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(x + 1) = ln x + ln 1

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given g(x) = e^x, find g(ln ln 5^2)

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = 3^x, find ƒ(log_3 (ln 3))

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log(x + 3) - log(2x) = [log(x + 3)/log(2x)]

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log_2 x, find ƒ(2^(log_2 2))

Work each problem. Which of the following is equivalent to ln(4x) - ln(2x) for x > 0? A. 2 ln x B. ln 2x C. (ln 4x)/(ln 2x) D. ln 2

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log6 [4(x + 1)] = log6 (4) + log6 (x + 1)

Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log￬2 [4 (x-3) ]

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. logb (x^3 + y^3) = 3 logb x + 3 logb y

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log(1000x)

Answer each of the following. Write log_3 12 in terms of natural logarithms using the change-of-base theorem.

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log4 (64/y)

Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 0.1

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log N^(-6)

Find each value. If applicable, give an approximation to four decimal places. See Example 1. log(387 * 23)

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log4 (√x/64)

Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 518 - log 342

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log √(100x)

For each substance, find the pH from the given hydronium ion concentration to the nearest tenth. See Example 2(a). crackers, 3.9*10^-9

Use a calculator to find an approximation to four decimal places for each logarithm. log₂/₃ 5/8

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log5 ∛((x^2 y)/24)

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Determine whether the wetland is a rich fen, a poor fen, or a bog. See Example 3. 2.49*10^-5

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log 5 + log 2

Solve each problem. Use a calculator to find an approximation for each logarithm. log 3.984

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log x + 3 log y

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln √e

In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln ∛(x/e)

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 5 ln x - 2 ln y

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln (27 * 943)

In Exercises 58–59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log4 0.863

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 3 ln x + 5 ln y - 6 ln z

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 98 - ln 13

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. (1/3) [2 ln(x + 5) - ln x - ln (x^2 - 4)]

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log14 87.5

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_8 0.59

Expand: log8((4√x)/(64y3))

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_√13 12

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log4 (2x^3) = 3 log4 (2x)

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(x + 1) = ln x + ln 1

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given g(x) = e^x, find g(ln 4)

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = 3^x, find ƒ(log_3 2)

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log(x + 3) - log(2x) = [log(x + 3)/log(2x)]

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log_2 x, find ƒ(2^7)

Work each problem. Which of the following is equivalent to 2 ln(3x) for x > 0? A. ln 9 + ln x B. ln 6x C. ln 6 + ln x D. ln 9x^2

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log6 [4(x + 1)] = log6 (4) + log6 (x + 1)

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. logb (x^3 + y^3) = 3 logb x + 3 logb y