# Solving Exponential Equations

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## Solving Exponential Equations Graphically

If the dependent variable of an exponential function written in the form $y = a \cdot b^x,$ is exchanged for a constant, say $C,$ the result is a one-variable equation: $C = a \cdot b^x.$

This type of equation is called an exponential equation, and can be solved graphically. This is done by first graphing the function $y = a \cdot b^x,$ then finding the $x$-coordinate of the point(s) on the graph with the $y$-coordinate $C.$ The $x$-coordinate(s) is the solution to the equation.## Solving Exponential Equations with the Same Base

There are different ways to algebraically solve exponential equations. If both sides of the equation can be written in the same base, equality can be used. For example, consider the equation $4^{2x}=64^2.$ Since $64=4^3,$ the equation can be written as follows. $\begin{aligned} 4^{2x} &=\left({\color{#0000FF}{4^3}} \right)^{2}\\ 4^{2x} &=4^6\\ \end{aligned}$ Now, we have two equivalent expressions with the same base. For the equality to hold, the exponents must also be equal. Thus,

$2x=6 \quad \Leftrightarrow \quad x=3.$## Solving an Exponential Equation with Logarithms

When the expressions on both sides of an exponential equation cannot be written with the same base, logarithms can used instead. Essentially, a logarithm is used to undo the exponent. Consider the equation $8^x=3.$ Since $x$ is the exponent of base $8,$ $\log_8$ can be applied to each side of the equation. This yields ${\color{#0000FF}{\log_8}}\left(8^x\right)={\color{#0000FF}{\log_8}}(3).$ Now, $\log_8\left(8^x\right)$ is equal to the exponent needed to make $8$ equal $8^x.$ This can also be expressed as $\log_8\left(8^x\right)=x.$ By using this to rewrite the left-hand side the equation becomes $x=\log_8(3).$ This is the exact solution to the equation. If a numerical value is required, use a calculator. However, the "log" button on a calculator corresponds to a common logarithm — $\log_{10}.$ When the logarithm has a base different than $10,$ the Change of Base rule can be used to express the logarithm with base $10.$ $\log_n(m)=\dfrac{\log(m)}{\log(n)}.$ For the example, $x$ equals

$\log_8(3)=\dfrac{\log_{10}(3)}{\log_{10}(8)}\approx 0.53.$## Exercises

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