# 5.5 Solving Exponential Equations and Logarithmic Equations

Solve exponential equations.

Solve logarithmic equations.

# Solving Exponential Equations

Equations with variables in the exponents, such as

are called **exponential equations**.

Sometimes, as is the case with the equation ${2}^{5x}=64$, we can write each side as a power of the same number:

We can then set the exponents equal and solve:

We use the following property to solve exponential equations.

This property follows from the fact that for any $a>0$, $a\ne 1$, $f(x)={a}^{x}$ is a one-to-one function. If ${a}^{x}={a}^{y}$, then $f(x)=f(y)$. Then since `f` is one-to-one, it follows that $x=y$. Conversely, if $x=y$, it follows ...

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