6. Exponential & Logarithmic Functions

Solve each equation. $5^{x^2 - 12} = 25^{2x}$

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

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 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

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. Give solutions in exact form. See Examples 5–9. log₈ (x + 2) + log₈ (x + 4) = log₈ 8

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

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. 5^2x + 3(5^x) = 28

Solve each equation. Give solutions in exact form. 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 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 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 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

n Exercises 92–93, rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. y = 73(2.6)^x

Use the formula for continuous compounding to solve Exercises 84–85. How long, to the nearest tenth of a year, will it take \$50,000 to triple in value at an annual rate of 7.5% compounded continuously?