6. Exponential and Logarithmic Functions

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 2^x=64

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 2^2x−1=32

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 6^(x−3)/4=√6

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 0.8^x = 4

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e^x=5.7

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. (1/3)^x = -3

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 5^x=17

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 5(1.015)^(x-1980) = 8

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 7^(x+2)=410

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 2e^2x + e^x = 6

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 3^2x+3^x−2=0

Solve each equation. Give solutions in exact form. See Examples 5–9. log_6 (2x + 4) = 2

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log3(x+4)=−3

Solve each equation. Give solutions in exact form. See Examples 5–9. ln(7 - x) + ln(1 - x) = ln (25 - x)

In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2^(4x-2) = 64

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log5 x+log5(4x−1)=1

Solve each equation. Give solutions in exact form. See Examples 5–9. ln(4x - 2) - ln 4 = -ln(x - 2)

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log(x+4)=log x+log 4

Solve each equation. Give solutions in exact form. See Examples 5–9. log x^2 = (log x)^2

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log x+log(x+3)=log 10

Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. y = A + B(1 - e^(-Cx)), for x

In Exercises 93–102, solve each equation. 2|ln x|−6=0

In Exercises 93–102, solve each equation. 5^(x^2−12)=25^2x

Write an equation for the inverse function of each one-to-one function given. ƒ(x) = (1/3)^x

Find ƒ^-1(x), and give the domain and range. ƒ(x) = e^(x-5)

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 5^x=125

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 3^1−x=1/27

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. (1/2)^x = 5

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 10^x=3.91

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. e^(3x-7) • e^-2x = 4e

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e^(5x−3) − 2=10,476

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. e^2x - 6e^x + 8 = 0

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e^4x+5e^2x−24=0

Solve each equation. Give solutions in exact form. See Examples 5–9. log(2 - x) = 0.5

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log2(x+25)=4

Solve each equation. Give solutions in exact form. See Examples 5–9. log(3x + 5) - log(2x + 4) = 0

In Exercises 60–63, 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. (log2 x)^4 = 4 log2 x

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ln√x+3=1

Solve each equation. Give solutions in exact form. See Examples 5–9. log(9x + 5) = 3 + log(x + 2)

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log2(x−6)+log2(x−4)−log2 x=2

Solve each equation. Give solutions in exact form. See Examples 5–9. . log_5 (x + 2) + log_5 (x - 2) = 1

In Exercises 74–79, solve each logarithmic equation. log4 (2x+1) = log4 (x-3) + log4 (x+5)

Solve each equation. Give solutions in exact form. See Examples 5–9. log_2 (log_2 x) = 1

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2 log x−log 7=log 112

Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. y = K/(1+ae^(-bx)), for b

In Exercises 93–102, solve each equation. 3^x+2 ⋅ 3^x=81

In Exercises 93–102, solve each equation. ln 3−ln(x+5)−ln x=0

To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)^(tn) and A = Pe^(rt) At what interest rate, to the nearest hundredth of a percent, will $16,000 grow to $20,000 if invested for 7.25 yr and interest is compounded quarterly?

Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 3^x

Find the error in the following 'proof' that 2 < 1.

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 9^x=27

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 3^x = 7

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. e^(x+1)=1/e

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. e^(x^2) = 100

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 2(1.05)^x + 3 = 10

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e^(1−5x)=793

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e^2x−3e^x+2=0

Solve each equation. Give solutions in exact form. See Examples 5–9. ln 4x = 1.5

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log4(x+5)=3

Solve each equation. Give solutions in exact form. See Examples 5–9. log_5 [(3x + 5)(x + 1)] = 1

Solve each equation. Give solutions in exact form. See Examples 5–9. log(x + 25) = log(x + 10) + log 4

In Exercises 60–63, 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 1) = 0

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 6+2 ln x=5

Solve each equation. Give solutions in exact form. See Examples 5–9. log x + log(x - 21) = log 100

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2 log3(x+4)=log3 9+2

In Exercises 74–79, solve each logarithmic equation. log2 (x+3) + log2 (x-3) =4

Solve each equation. Give solutions in exact form. See Examples 5–9. ln e^x - 2 ln e = ln e^4

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log(x+4)−log 2=log(5x+1)

Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. I = E/R (1- e^(-(Rt)/2), for t

In Exercises 93–102, solve each equation. 5^2x ⋅ 5^4x=125

In Exercises 93–102, solve each equation. ln(2x+1)+ln(x−3)−2 ln x=0

To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)^(tn) and A = Pe^(rt) Find t, to the nearest hundredth of a year, if $1786 becomes $2063 at 2.6%, with interest compounded monthly.

Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬4 (x+3), g(x) = 4^x + 3

Find ƒ^-1(x), and give the domain and range. ƒ(x) = 2 ln 3x

Exercises 137–139 will help you prepare for the material covered in the next section. Solve: (x + 2)/(4x + 3) = 1/x

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 32^x=8

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 8^(x+3)=16^(x−1)

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 6^(x+1) = 4^(2x-1)

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 3(2)^(x-2) + 1 = 100

Graph f(x) = 2^x and g(x) = log2 x in the same rectangular coordinate system. Use the graphs to determine each function's domain and range.

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 3e^5x=1977

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 5^(2x+3)=3^(x−1)

Solve each equation. Give solutions in exact form. See Examples 5–9. 5 ln x = 10

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ln x=2

Solve each equation. Give solutions in exact form. See Examples 5–9. ln x + ln x^2 = 3

Solve each equation. Give solutions in exact form. See Examples 5–9. log_2 [(2x + 8)(x + 4)] = 5

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 5 ln(2x)=20

Solve each equation. Give solutions in exact form. See Examples 5–9. log_2 (x^2 - 100) - log_2 (x + 10) = 1

In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 8^x = 12143

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log2(x+2)−log2(x−5)=3

Solve each equation. Give solutions in exact form. See Examples 5–9. log_2 (2x - 3) + log_2 (x + 1) = 1

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2 log x=log 25

Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. r = p - k ln t, for t

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ln(x−2)−ln(x+3)=ln(x−1)−ln(x+7)

In Exercises 93–102, solve each equation. 3^x2=45

Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. A = P (1 + r/n)^(tn), for t

Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬2 x+1, g(x) = 2^x-1

Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 4^x+2

Find ƒ^-1(x), and give the domain and range. ƒ(x) = e^(x+1) - 4

Exercises 137–139 will help you prepare for the material covered in the next section. Solve: x(x - 7) = 3.

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 4^2x−1=64

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 4^x=1/√2

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 4^(x-1) = 3^2x

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 0.05(1.15)^x = 5

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 5e^x=23

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 7^0.3x=813

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 5^2x + 3(5^x) = 28

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log3 x=4

Solve each equation. Give solutions in exact form. See Examples 5–9. log_4 (x^3 + 37) = 3

Solve each equation. Give solutions in exact form. See Examples 5–9. log_3 [(x + 5)(x - 3)] = 2

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log4(3x+2)=3

Solve each equation. Give solutions in exact form. See Examples 5–9. log_8 (x + 2) + log_8 (x + 4) = log_8 8

In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 10^x = 7000

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log3(x+6)+log3(x+4)=1

Solve each equation. Give solutions in exact form. See Examples 5–9. ln(5 + 4x) - ln(3 + x) = ln 3

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log(3x−3)=log(x+1)+log 4

Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. p = a + (k/ln x), for x

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ln(x−4)+ln(x+1)=ln(x−8)

Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. log A = log B - C log x, for A

In Exercises 93–102, solve each equation. 3|log x|−6=0

Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = 5^x, g(x) = log￬5 x

Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 5^x + 1

Find ƒ^-1(x), and give the domain and range. ƒ(x) = e^x + 10

Exercises 137–139 will help you prepare for the material covered in the next section. Solve for x: a(x - 2) = b(2x + 3)