Solve Exponential Equations Using Logs

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Key Ideas Math Facts Calculator Easy Medium Hard

What to Remember

When the variable is in the exponent

Free the variable by converting to a log.

10^x = 6

Change to a log with the same base.

log₁₀ 6 = x

When the base is e

Free the x by converting to a log.

e^x = 5

log_e 5 = x

ln 5 = x

A shifted exponent

10^x+3 = 11

Change to a log with the same base.

log₁₀ 11 = x+3

x = log₁₀ 11 − 3

Same base on both sides

Divide the powers of 10 (or of e), then take log or ln.

10^2x = 7 · 10^x

10^x = 7

Change to a log with the same base.

log₁₀ 7 = x

Math Facts

When a question writes log with no base, that means log₁₀.

ln means log_e, the natural logarithm.

A different base is written as a subscript, for example log₂ 8. This is read as log base 2 of 8.

TI-84 Plus CE: LOG and LN

Press LOG for base 10. Press LN for natural log.
Type the number, then close the parenthesis and press ENTER.
Round the display to 3 significant figures unless the question asks for an exact answer.

Exact answers stay in log or ln form, for example \(x = \log 6\) or \(x = \ln 5\). Do not switch to a decimal unless the question says to use technology.

Level 1

Easy

One exponential equal to a number. Leave exact answers in log or ln form. Calculator items: 3 significant figures.

Example

Solve for x. Give an exact answer. 10^x = 6

Free the variable by converting to a log.

10^x = 6

Change to a log with the same base.

log₁₀ 6 = x

Answer: log₁₀ 6

Solve for x. Give an exact answer. 10^x = 14

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10^x = 14

Change to a log with the same base.

log₁₀ 14 = x

Answer: log₁₀ 14

Solve for x. Give an exact answer. 10^x = 0.4

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10^x = 0.4

Change to a log with the same base.

log₁₀ 0.4 = x

Answer: log₁₀ 0.4

Solve for x. Give an exact answer. e^x = 5

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Free the x by converting to a log.

e^x = 5

log_e 5 = x

ln 5 = x

Answer: ln 5

Solve for x. Give an exact answer. e^x = 12

Show Answer

Free the x by converting to a log.

e^x = 12

log_e 12 = x

ln 12 = x

Answer: ln 12

Use technology. Evaluate to 3 significant figures. log 87.4

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Press LOG, type 87.4, ENTER.

log₁₀ 87.4 = 1.94 (3 s.f.)

Answer: 1.94

Use technology. Evaluate to 3 significant figures. ln 52.6

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Press LN, type 52.6, ENTER.

ln 52.6 = 3.96 (3 s.f.)

Answer: 3.96

Level 2

Medium

The exponent is x plus or minus a number. Isolate the exponential, then take log or ln.

Example

Solve for x. Give an exact answer. 10^x+3 = 11

10^x+3 = 11

Change to a log with the same base.

log₁₀ 11 = x+3

x = log₁₀ 11 − 3

Answer: log₁₀ 11 − 3

Solve for x. Give an exact answer. 10^x−1 = 9

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10^x−1 = 9

Change to a log with the same base.

log₁₀ 9 = x−1

x = log₁₀ 9 + 1

Answer: log₁₀ 9 + 1

Solve for x. Give an exact answer. 10^x+5 = 40

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10^x+5 = 40

Change to a log with the same base.

log₁₀ 40 = x+5

x = log₁₀ 40 − 5

Answer: log₁₀ 40 − 5

Solve for x. Give an exact answer. e^x−4 = 8

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e^x−4 = 8

Free the x by converting to a log.

log_e 8 = x−4

ln 8 = x−4

x = ln 8 + 4

Answer: ln 8 + 4

Solve for x. Give an exact answer. e^x+1 = k+3

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e^x+1 = k+3

Free the x by converting to a log.

log_e (k+3) = x+1

ln(k+3) = x+1

x = ln(k+3) − 1

Answer: ln(k+3) − 1

Solve for x. Give an exact answer. 10^x−3 = 200

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10^x−3 = 200

Change to a log with the same base.

log₁₀ 200 = x−3

x = log₁₀ 200 + 3

Answer: log₁₀ 200 + 3

Level 3

Hard

Exponential terms on both sides. Divide first so one power remains, then take log or ln.

Example

Solve for x. Give an exact answer. 10^2x = 7 · 10^x

10^2x10^x = 7

10^x = 7

Change to a log with the same base.

log₁₀ 7 = x

Answer: log₁₀ 7

Solve for x. Give an exact answer. 10^3x = 2 · 10^x

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10^2x = 2

Change to a log with the same base.

log₁₀ 2 = 2x

x = log₁₀ 22

Answer: log₁₀ 22

Solve for x. Give an exact answer. e^2x = 5e^1−x

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2x = ln 5 + 1 − x

3x = ln 5 + 1

x = ln 5 + 13

Answer: ln 5 + 13

Solve for x. Give an exact answer. e^x+2 = 4e^3−x

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x+2 = ln 4 + 3 − x

2x = ln 4 + 1

x = ln 4 + 12

Answer: ln 4 + 12

Solve for x. Give an exact answer. 5 · 10^x = 20

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10^x = 4

Change to a log with the same base.

log₁₀ 4 = x

Answer: log₁₀ 4

Solve for x. Give an exact answer. e^x+3 = 2k−1

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e^x+3 = 2k−1

Free the x by converting to a log.

log_e (2k−1) = x+3

ln(2k−1) = x+3

x = ln(2k−1) − 3

Answer: ln(2k−1) − 3

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