Exponential Equations with Unlike Bases

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Key Ideas Easy Medium Hard

If b^x = N, then x = log_b N.

log_b N = ln N ÷ ln b.

What to Remember

Bases that do not match

If you cannot write both sides as powers of one base, get a single power equal to a number.

Then the exponent is a log.

The division step

From 8 · 4^x = 7^x, divide both sides by 4^x.

8 = 74^x. So x = log of 8 with that base.

A log is an exact answer

x = log_b N is exact. A calculator can turn that into a decimal later.

log_b N = ln N ÷ ln b.

Level 1

Easy

One side is a number

Example

Solve 5^x = 12.

If b^x = N, then x = log_b N.

5^x = 12

\(x = \log_{5} 12\)

Answer: \(x = \log_{5} 12\)

Solve 3^x = 10.

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The bases do not match, so write a log.

If b^x = N, then x = log_b N.

3^x = 10

\(x = \log_{3} 10\)

Answer: \(x = \log_{3} 10\)

Solve 2^x = 7.

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The bases do not match, so write a log.

If b^x = N, then x = log_b N.

2^x = 7

\(x = \log_{2} 7\)

Answer: \(x = \log_{2} 7\)

Solve 6^x = 5.

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The bases do not match, so write a log.

If b^x = N, then x = log_b N.

6^x = 5

\(x = \log_{6} 5\)

Answer: \(x = \log_{6} 5\)

Solve e^x = 9.

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If e^x = N, then x = ln N.

e^x = 9

\(x = \ln 9\)

Answer: \(x = \ln 9\)

Solve 10^{x − 2} = 4.

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Write the power as a log.

x − 2 = log 4

Add 2.

\(x = \log 4 + 2\)

Answer: \(x = \log 4 + 2\)

Level 2

Medium

The variable is in both exponents

Example

Solve 2^{x + 3} = 7^x.

2^{x + 3} = 7^x

Split the left side.

2^x · 2³ = 7^x

2^x · 8 = 7^x

Divide both sides by 2^x.

8 = 72^x

If b^x = N, then x = log_b N.

\(x = \log_{\frac{7}{2}} 8\)

Answer: \(x = \log_{\frac{7}{2}} 8\)

Solve 6^x = 5^{x + 2}.

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6^x = 5^x · 25

Divide both sides by 5^x.

65^x = 25

If b^x = N, then x = log_b N.

\(x = \log_{\frac{6}{5}} 25\)

Answer: \(x = \log_{\frac{6}{5}} 25\)

Solve 2^3x = 5^{x + 1}.

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8^x = 5^x · 5

Divide both sides by 5^x.

85^x = 5

\(x = \log_{\frac{8}{5}} 5\)

Answer: \(x = \log_{\frac{8}{5}} 5\)

Solve 3^{x + 2} = 7^x.

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3^x · 9 = 7^x

Divide both sides by 3^x.

9 = 73^x

\(x = \log_{\frac{7}{3}} 9\)

Answer: \(x = \log_{\frac{7}{3}} 9\)

Solve 7^2x = 3^{x + 1}.

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49^x = 3^x · 3

Divide both sides by 3^x.

493^x = 3

\(x = \log_{\frac{49}{3}} 3\)

Answer: \(x = \log_{\frac{49}{3}} 3\)

Solve 5^{x + 2} = 3^2x.

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5^x · 25 = 9^x

Divide both sides by 5^x.

25 = 95^x

\(x = \log_{\frac{9}{5}} 25\)

Answer: \(x = \log_{\frac{9}{5}} 25\)

Level 3

Hard

Rewrite a power before you divide

Example

Solve 27^{x − 1} = 4^x.

27^{x − 1} = 4^x

27 = 3³, so 27^{x − 1} = 3^{3x − 3}

3^{3x − 3} = 27^x · 3⁻³

127 · 27^x = 4^x

Divide both sides by 4^x and multiply by 27.

274^x = 27

\(x = \log_{\frac{27}{4}} 27\)

Answer: \(x = \log_{\frac{27}{4}} 27\)

Solve 9^{x + 1} = 2^4x.

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9^x · 9 = 16^x

Divide both sides by 9^x.

9 = 169^x

\(x = \log_{\frac{16}{9}} 9\)

Answer: \(x = \log_{\frac{16}{9}} 9\)

Solve 25^{x − 1} = 4^{x + 1}.

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25^x · 125 = 4^x · 4

Multiply both sides by 25 and divide by 4^x.

254^x = 100

\(x = \log_{\frac{25}{4}} 100\)

Answer: \(x = \log_{\frac{25}{4}} 100\)

Solve 8^{x + 2} = 5^3x.

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8 = 2³ and 125 = 5³

8^x · 64 = 125^x

Divide both sides by 8^x.

64 = 1258^x

\(x = \log_{\frac{125}{8}} 64\)

Answer: \(x = \log_{\frac{125}{8}} 64\)

Solve 4^{2x − 1} = 3^{x + 2}.

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4 = 2², so the left side is 2^{4x − 2}

16^x · 14 = 3^x · 9

Multiply by 4 and divide by 3^x.

163^x = 36

\(x = \log_{\frac{16}{3}} 36\)

Answer: \(x = \log_{\frac{16}{3}} 36\)

Solve 7^{2x + 1} = 3^{x − 2}.

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49^x · 7 = 3^x · 19

Divide by 3^x and divide by 7.

493^x = 163

\(x = \log_{\frac{49}{3}} \frac{1}{63}\)

Answer: \(x = \log_{\frac{49}{3}} \frac{1}{63}\)

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