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**Solving Exponential Equations Using Logarithms**

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**Solving Exponential Functions with Logarithms**

There are three main steps to solving exponential functions using logarithms. Express each side of the equation as a single term. Take the logarithm of both sides, choosing a convenient base. Solve the resulting equation.

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Example Solve the equation 4 * 2x-3 = 5 * 3x.

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Solution: Step 1 Conveniently, each side is already expressed as a single term. We can simplify the left side, though, by noticing that 4 = 22, so 4 * 2x-3 is equal to 2x-1. This simplifies our equation to 2x-1 = 5 * 3x

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**Solution: Step 2 Now we take the logarithm of both sides.**

Each side has a different base, so there’s no truly convenient base for us to use. We’ll use the natural log. ln(2x-1) = ln(5 * 3x) Now we can solve the equation using the properties of logarithms.

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**Solution: Step 3 ln(2x-1) = ln(5 * 3x) (x-1)ln(2) = ln(5) + xln(3)**

(ln(2) – ln(3))x = ln 5 + ln 2 Using a calculator, we can calculate all of these natural logs.

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**Solution: Part 3 ln 2 = 0.69 ln 3 = 1.10 ln 5 = 1.61**

Thus, -0.41x = 2.30 x = -5.6

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