6. Exponential and Logarithmic Functions

In Exercises 1–8, write each equation in its equivalent exponential form. 4 = log₂ 16

In Exercises 1–8, write each equation in its equivalent exponential form. 2 = log9 x

In Exercises 9–20, write each equation in its equivalent logarithmic form. 2^-4 = 1/16

In Exercises 13–15, write each equation in its equivalent exponential form. 1/2 = log49 7

In Exercises 9–20, write each equation in its equivalent logarithmic form. 13^2 = x

If the statement is in exponential form, write it in an equivalent logarithmic form. If the statement is in logarithmic form, write it in exponential form. log￬5 5 = 1

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log16 4

In Exercises 21–42, evaluate each expression without using a calculator. log2 64

Solve each equation. x = log￬7 ⁵√7

In Exercises 21–42, evaluate each expression without using a calculator. log5 (1/5)

In Exercises 21–42, evaluate each expression without using a calculator. log64 8

Solve each equation. x = log￬4 ∛16

In Exercises 32–35, the graph of a logarithmic function is given. Select the function for each graph from the following options: f(x) = logx, g(x) = log(-x), h(x) = log(2-x), r(x)= 1+log(2-x)

In Exercises 21–42, evaluate each expression without using a calculator. log4 1

In Exercises 43– 48, match the function with its graph from choices A–F. ƒ(x) = log￬2 x

Graph each function. ƒ(x) = log￬6 (x-2)

In Exercises 53-58, begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = (1/2)log₂ x

The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range.

g(x) = 1-log x

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 function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. h(x) = ln(x/2)

The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. g(x) = 2 ln x

In Exercises 75–80, find the domain of each logarithmic function. f(x) = log (2 - x)

In Exercises 81–100, evaluate or simplify each expression without using a calculator. log 100

In Exercises 81–100, evaluate or simplify each expression without using a calculator. In e

In Exercises 81–100, evaluate or simplify each expression without using a calculator. In (1/e^6)

In Exercises 81–100, evaluate or simplify each expression without using a calculator. 10^(log √x)

In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. log3 (x-1) = 2

In Exercises 105–108, evaluate each expression without using a calculator. log2 (log3 81)

In Exercises 109–112, find the domain of each logarithmic function. f(x) = ln (x² - x − 2)

145. Without using a calculator, determine which is the greater number: log4 60 or log3 40.

In Exercises 1–8, write each equation in its equivalent exponential form. 2 = log3 x

In Exercises 9–20, write each equation in its equivalent logarithmic form. 5^4 = 625

If the statement is in exponential form, write it in an equivalent logarithmic form. If the statement is in logarithmic form, write it in exponential form. 3^4 = 81

In Exercises 9–20, write each equation in its equivalent logarithmic form. ∛8 = 2

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log5 (1/5)

In Exercises 21–42, evaluate each expression without using a calculator. log4 16

Solve each equation. x = log￬8 ∜8

In Exercises 21–42, evaluate each expression without using a calculator. log3 27

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. ln e^5

In Exercises 21–42, evaluate each expression without using a calculator. log2 (1/√2)

Solve each equation. log￬2 x = 3

In Exercises 21–42, evaluate each expression without using a calculator. log5 5

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = ln x and g(x) = - ln (2x)

Graph each function. ƒ(x) = log￬10 x

In Exercises 53-58, begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. h(x)=1+ log₂ x

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 function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. g(x) = ln (x+2)

In Exercises 75–80, find the domain of each logarithmic function. f(x) = log5 (x+4)

In Exercises 75–80, find the domain of each logarithmic function. f(x) = ln (x-2)²

In Exercises 81–100, evaluate or simplify each expression without using a calculator. In 1

In Exercises 81–100, evaluate or simplify each expression without using a calculator. In e^6

In Exercises 81–100, evaluate or simplify each expression without using a calculator. e^(ln 5x^2)

In Exercises 81–100, evaluate or simplify each expression without using a calculator. 10^(log ∛x)

Use properties of logarithms to rewrite each function, then graph. ƒ(x) = log￬3 x+1/9

In Exercises 105–108, evaluate each expression without using a calculator. log5 (log2 32)

In Exercises 105–108, evaluate each expression without using a calculator. log (ln e)

Without using a calculator, find the exact value of log4 [log 3 (log₂ 8)].

In Exercises 1–8, write each equation in its equivalent exponential form. log6 216 = y

In Exercises 16–18, write each equation in its equivalent logarithmic form. 13^y = 874

In Exercises 9–20, write each equation in its equivalent logarithmic form. 7^y = 200

Solve each equation. log￬x 27/64 = 3

In Exercises 21–42, evaluate each expression without using a calculator. log2 64

In Exercises 21–42, evaluate each expression without using a calculator. log7 √7

Solve each equation. log￬4 x = 3

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = log x and g(x) = - log (x+3)

In Exercises 21–42, evaluate each expression without using a calculator. 8^(log8 19)

Solve each equation. 3x - 15 = log￬x 1 (x>0, x≠1)

In Exercises 53-58, begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log₂ (x + 1)

The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. h(x) = log x − 1

Graph each function. Give the domain and range. ƒ(x) = | log￬2 (x+3) |

In Exercises 81–100, evaluate or simplify each expression without using a calculator. 10^(log 33)

In Exercises 81–100, evaluate or simplify each expression without using a calculator. In e^9x

Given that log￬10 2 ≈ 0.3010 and log￬10 3 ≈ 0.4771, find each logarithm without using a calculator. log￬10 √30

In Exercises 105–108, evaluate each expression without using a calculator. log5 (log7 7)

Without using a calculator, find the exact value of: [log3 81 - log𝝅 1]/[log2√2 8 - log 0.001]

In Exercises 1–8, write each equation in its equivalent exponential form. 5= logb 32

In Exercises 13–15, write each equation in its equivalent exponential form. log3 81 = y

In Exercises 9–20, write each equation in its equivalent logarithmic form. b^3 = 1000

Solve each equation. x = log￬3 1/81

In Exercises 21–42, evaluate each expression without using a calculator. log4 16

In Exercises 21–42, evaluate each expression without using a calculator. log2 (1/8)

Solve each equation. log￬x 25 = -2

In Exercises 36–38, begin by graphing f(x) = log2 x Then use transformations of this graph to graph the given function. What is the graph's x-intercept? What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log2 (x-2)

Solve each equation. log￬1/3 (x+6) = -2

In Exercises 21–42, evaluate each expression without using a calculator. log5 5^7

Graph f(x) = (1/2)^x and g(x) = log(1/2) x in the same rectangular coordinate system.

The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. g(x) = log(x − 1)

Graph each function. Give the domain and range. ƒ(x) = (log￬2 x) + 3

In Exercises 81–100, evaluate or simplify each expression without using a calculator. log 10^7

In Exercises 81–100, evaluate or simplify each expression without using a calculator. e^ln 125

Given that log￬10 2 ≈ 0.3010 and log￬10 3 ≈ 0.4771, find each logarithm without using a calculator. log￬10 9/4

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 1–8, write each equation in its equivalent exponential form. log6 216 = y

In Exercises 9–20, write each equation in its equivalent logarithmic form. 13^2 = x

If the statement is in exponential form, write it in an equivalent logarithmic form. If the statement is in logarithmic form, write it in exponential form. log￬4 1/64 = -3

In Exercises 16–18, write each equation in its equivalent logarithmic form. 13^y = 874

In Exercises 21–42, evaluate each expression without using a calculator. log5 (1/5)

Solve each equation. x = 2^log2 9

In Exercises 32–35, the graph of a logarithmic function is given. Select the function for each graph from the following options: f(x) = logx, g(x) = log(-x), h(x) = log(2-x), r(x)= 1+log(2-x)

Solve each equation. log￬1/2 (x+3) = -4

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = log x and g(x) = - log (x+3)

Graph f(x) = 4^x and g(x) = log4 x in the same rectangular coordinate system.

In Exercises 53-58, begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. h(x) = 2 + log2 x

Graph each function. ƒ(x) = log￬1/2 (x+3) - 2

In Exercises 81–100, evaluate or simplify each expression without using a calculator. 10^(log 33)

Given that log￬10 2 ≈ 0.3010 and log￬10 3 ≈ 0.4771, find each logarithm without using a calculator. log￬10 3/2

In Exercises 81–100, evaluate or simplify each expression without using a calculator. In e^9x

In Exercises 105–108, evaluate each expression without using a calculator. log5 (log7 7)

Without using a calculator, find the exact value of: [log3 81 - log𝝅 1]/[log2√2 8 - log 0.001]

In Exercises 1–8, write each equation in its equivalent exponential form. 5= logb 32

In Exercises 13–15, write each equation in its equivalent exponential form. log3 81 = y

If the statement is in exponential form, write it in an equivalent logarithmic form. If the statement is in logarithmic form, write it in exponential form. log￬√3 81 = 8

In Exercises 21–42, evaluate each expression without using a calculator. log3 27

Solve each equation. x = 3^log3 8

In Exercises 21–42, evaluate each expression without using a calculator. log5 5

Solve each equation. log￬9 x = 5/2

In Exercises 36–38, begin by graphing f(x) = log2 x Then use transformations of this graph to graph the given function. What is the graph's x-intercept? What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log2 (x-2)

Graph each function. ƒ(x) = log￬3 (x-1) + 2

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 function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range.

h(x) = ln (2x)

In Exercises 81–100, evaluate or simplify each expression without using a calculator. log 10^7

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

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)]