Section 4.5 - Exponential and Logarithmic Equations

Section Objectives

Solve exponential equations.
Solve logarithmic equations.
Use and solve exponential and logarithmic equations in applications.

Two Important Properties for Solving Equations

Some kinds of logarithmic and exponential equations can be solved by appealing to the one-to-one property of logarithms and exponentials:

a^x = a^y \quad \mbox{if and only if} \quad x=y,\\[0.1 in] \log_a x = \log_a y \quad \mbox{if and only if} \quad x = y.

In order to use these properties, (1) the equation must be written with a single term on each side of the equation, and (2) each side must have an expression in the same base.

Examples

$x$ $\quad 3^{x-1}=27$ (Solution)

$x$ $\quad \log_2 (3x-4)= \log_2(x+2)$ (Solution)

$t$ $\quad \displaystyle 25^{4-t}=\left( \frac{1}{5} \right)^{3t+1}$ (Solution)

Other equations can be solved by using the properties of logarithms and exponentials. The following examples are typical.

$x$ $\quad 7^x=60$ (Solution)

$x$ $\quad 18000e^{-4t} -500 = 1500$ (Solution)

$x$ $\quad 4^{2x-7}=5^{3x+1}$ (Solution)

$x$ $\quad \ln(x-4) = \ln(x+6)-\ln x$ (Solution)

$t$ $\quad 4 \log_3 (2t-7) = 8$ (Solution)

For further help...

Do a Google search for "exponential equations" and/or "logarithmic equations".
Use WolframAlpha to solve your equation. (You can even see the steps.)