# Properties of Logarithms - Video Tutorials & Practice Problems

## Product, Quotient, and Power Rules of Logs

## Expand & Condense Log Expressions

Write the log expression as a single log.

${\mathrm{log}}_{2}\frac{1}{9x}+2{\mathrm{log}}_{2}3x$

${\mathrm{log}}_{2}x$

${\mathrm{log}}_{2}\frac{1}{3x}$

${\mathrm{log}}_{2}1$

${\mathrm{log}}_{2}3x$

Write the log expression as a single log.

$\mathrm{ln}\frac{3x}{y}+2\mathrm{ln}2y-\mathrm{ln}4x$

$\mathrm{ln}\frac{3xy}{4}$

$\mathrm{ln}\left(12{x}^{2}\right)$

$\mathrm{ln}\left(\frac{3}{2}\right)$

$\mathrm{ln}\left(3y\right)$

Write the single logarithm as a sum or difference of logs.

$\log_3\left(\frac{\sqrt{x}}{9y^2}\right)$

$2{\mathrm{log}}_{3}x-2-{\mathrm{log}}_{3}9y$

$\frac{1}{2}{\mathrm{log}}_{3}x-2-2{\mathrm{log}}_{3}y$

$\frac{1}{2}{\mathrm{log}}_{3}x+2{\mathrm{log}}_{3}3y$

$\frac{1}{2}{\mathrm{log}}_{3}x-2{\mathrm{log}}_{3}9y$

Write the single logarithm as a sum or difference of logs.

${\mathrm{log}}_{5}\left(\frac{5{(2x+3)}^{2}}{{x}^{3}}\right)$

$5+2{\mathrm{log}}_{5}(2x+3)+{\mathrm{log}}_{5}3x$

$2{\mathrm{log}}_{5}(2x+3)+3{\mathrm{log}}_{5}x$

$1+2{\mathrm{log}}_{5}(2x+3)+3{\mathrm{log}}_{5}x$

${\mathrm{log}}_{5}(2x+3)+{\mathrm{log}}_{5}x$

## Change of Base Property

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.

${\mathrm{log}}_{3}17$

0.39

2.58

1.23

0.48

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.

${\mathrm{log}}_{9}67$

1.91

0.52

0.95

1.83

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log.

${\mathrm{log}}_{8}41$

1.61

0.9

0.56

1.79

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log.

${\mathrm{log}}_{2}3789$

0.08

11.89

3.58

0.30

## Do you want more practice?

- In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Whe...
- In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Whe...
- In Exercises 1–8, write each equation in its equivalent exponential form. 5= logb 32
- In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Whe...
- Answer each of the following. Write log_3 12 in terms of natural logarithms using the change-of-base theorem.
- Answer each of the following. Between what two consecutive integers must log_2 12 lie?
- Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 10^12
- Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 0.1
- In Exercises 13–15, write each equation in its equivalent exponential form. log3 81 = y
- Find each value. If applicable, give an approximation to four decimal places. See Example 1. . log 63
- If the statement is in exponential form, write it in an equivalent logarithmic form. If the statement is in lo...
- Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 0.0022
- Find each value. If applicable, give an approximation to four decimal places. See Example 1. log(387 * 23)
- Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 518/342
- Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 387 + log 23
- In Exercises 21–42, evaluate each expression without using a calculator. log3 27
- Solve each equation. x = 3^log3 8
- Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 518 - log 342
- For each substance, find the pH from the given hydronium ion concentration to the nearest tenth. See Example 2...
- For each substance, find the pH from the given hydronium ion concentration to the nearest tenth. See Example 2...
- Use a calculator to find an approximation to four decimal places for each logarithm. ln 144,000
- For each substance, find the pH from the given hydronium ion concentration to the nearest tenth. See Example 2...
- Find the [H_3O^+] for each substance with the given pH. Write answers in scientific notation to the nearest te...
- Use a calculator to find an approximation to four decimal places for each logarithm. log₂/₃ 5/8
- In Exercises 21–42, evaluate each expression without using a calculator. log5 5
- Solve each equation. log￬9 x = 5/2
- Find the [H_3O^+] for each substance with the given pH. Write answers in scientific notation to the nearest te...
- In Exercises 36–38, begin by graphing f(x) = log2 x Then use transformations of this graph to graph the given ...
- Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Det...
- Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Det...
- Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Det...
- In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression...
- In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression...
- 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...
- Solve each problem. Use a calculator to find an approximation for each logarithm. log 3.984
- Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln e^1.6
- Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 1/e^2
- 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. Wh...
- Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 28
- Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 0.00013
- In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Wh...
- In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Wh...
- Graph each function. ƒ(x) = log￬3 (x-1) + 2
- In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression...
- Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln (27 * 943)
- In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression...
- Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 98/13
- Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 27 + ln 943
- In Exercises 58–59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal p...
- In Exercises 58–59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal p...
- Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 98 - ln 13
- Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 84 - ln 17
- Graph each function. Give the domain and range. ƒ(x) = | log￬1/2 (x-2) |
- The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each...
- In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal p...
- In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal p...
- In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal p...
- Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example...
- 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 (...
- Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example...
- Expand: log8((4√x)/(64y3))
- Expand: log8((4√x)/(64y3))
- In Exercises 81–100, evaluate or simplify each expression without using a calculator. log 10^7
- Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example...
- 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)
- 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.
- Solve: log₂ (x+9) — log₂ x = 1.
- 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 yo...
- In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support yo...
- In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support yo...
- Given that log￬10 2 ≈ 0.3010 and log￬10 3 ≈ 0.4771, find each logarithm without using a calculator. log￬10 6
- In Exercises 81–100, evaluate or simplify each expression without using a calculator. e^ln 125
- 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...
- Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(...
- Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(...
- Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(...
- Retaining the Concepts. Expand: log7 (5√x/49y^10) fifth root of x
- 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. l...
- 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....
- Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log￬2 [4 (x-3) ]
- Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log￬3 [9 (x+2) ]
- In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. log4 x=-3
- In Exercises 109–112, find the domain of each logarithmic function. f(x) = log[(x+1)/(x-5)]
- In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the n...
- In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the n...
- In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the n...
- If log 3 = A and log 7 = B, find log7 (9) in terms of A and B.