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What Are Exponential Equations?

An exponential equation is an equation where x is in the exponent. To get x down from the exponent, you have to use logarithms. You are free to use either log ⁡ or ln ⁡ .

When working with exponential and logarithmic equations it is important to remember the logarithmic and exponential rules! You solve these types of equations like this:

Rule

Exponential Equations

The same rules are in place for both ln ⁡ and log ⁡ . You are most likely to use ln ⁡ .

ax = b ln ⁡ ax = ln ⁡ b

Do the logarithm on both sides to get x down:

x ln ⁡ a = ln ⁡ b

Divide by ln ⁡ a on both sides:

x = ln ⁡ b ln ⁡ a,a,b > 0

Note! The answer will be the same whether you use ln ⁡ or log ⁡ !

Example 1

Solve the equation 10x = 3000

10x = 3000 ln ⁡ 10x = ln ⁡ 3000 x = ln ⁡ 3000 ln ⁡ 10 x ≈3.477

Example 2

Solve the equation 32 ⋅ 2x = 128

32 ⋅ 2x = 128| ÷ 32 2x = 4 ln ⁡ 2x = ln ⁡ 4 x ln ⁡ 2 = ln ⁡ 4 x = ln ⁡ 4 ln ⁡ 2 = ln ⁡ 22 ln ⁡ 2 = 2 ⋅ ln ⁡ 2 ln ⁡ 2 = 2

Example 3

Solve the equation 5 ⋅ 4x = 45

5 ⋅ 4x = 45| ÷ 5 4x = 9 ln ⁡ 4x = ln ⁡ 9 x ln ⁡ 4 = ln ⁡ 9| ÷ ln ⁡ 4 x = ln ⁡ 9 ln ⁡ 4 = ln ⁡ 32 ln ⁡ 22 = 2 ln ⁡ 3 2 ln ⁡ 2 = ln ⁡ 3 ln ⁡ 2

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What Are Logarithmic Equations?
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