# Section 4.5 - Exponential and Logarithmic Equations

Section Objectives

1. Solve exponential equations.
2. Solve logarithmic equations.
3. 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:

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

• Solve for $x$: $\quad 3^{x-1}=27$

• Solve for $x$: $\quad \log_2 (3x-4)= \log_2(x+2)$

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

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

• Solve for $x$: $\quad 7^x=60$

• Solve for $x$: $\quad 18000e^{-04t} -500 = 1500$

• Solve for $x$: $\quad 4^{2x-7}=5^{3x+1}$

• Solve for $x$: $\quad \ln(x-4) = \ln(x+6)-\ln x$

• Solve for $t$: $\quad 4 \log_3 (2t-7) = 8$