Solving exponential equations with base-10 logs | Year 11

01 / LEARN

The idea, step by step

Isolate the exponential term

When the unknown is in an exponent and equal bases are not convenient, first isolate the powered expression. Then take a common log of both positive sides and move the exponent out using the power law.

The resulting equation is usually linear in the unknown. In the formula below, require $b > 0$ , $b \neq 1$ , $k > 0$ and $m \neq 0$ so that both logarithms and the divisions are valid. Keep the calculator expression unrounded until the final answer.

b^{m x + c} = k ⟹ x = \frac{log k / log b - c}{m}

Interpret a time solution

A real-valued solution describes a crossing time in a continuous model. If the process is only checked at completed whole periods, examine the integers on either side and identify the first period satisfying the original condition.

For decay factors between zero and one, the logarithm of the factor is negative. If you divide an inequality by that negative number, reverse the inequality sign. A numerical substitution is a useful check.

A common trap

Rounding each logarithm to two decimal places before division.

Instead: Evaluate the complete log ratio with full calculator precision, then round the final value.

02 / WORKED EXAMPLES

See the thinking

Example 1

A linear expression in an exponent

Solve $3^{2 x - 1} = 20$ , to four decimal places.

Take logs: $(2 x - 1) log 3 = log 20$ .
$2 x - 1 = log 20 / log 3$ .
$x = (1 + log 20 / log 3) / 2 \approx 1.8634$ .

x approximately 1.8634

Example 2

Isolate a decay term

Solve $5 - 2 {(0.7)}^{t} = 4$ for t.

Rearrange: $2 {(0.7)}^{t} = 1$ , so ${(0.7)}^{t} = 0.5$ .
$t = log 0.5 / log 0.7 \approx 1.9434$ .

t approximately 1.9434

Check your understanding

Solve $4^{x} = 9$ using base-ten logs.

Show answer

$x = log 9 / log 4 \approx 1.5850$

Apply the power law after taking common logs.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Solve $8^{x} = 12$ to four decimal places.
2 marks
Solve $2^{x + 3} = 7$ to four decimal places.
2 marks
Solve $4 {(1.2)}^{t} = 10$ to three decimal places.
2 marks
A 600-unit quantity grows by 8% at the end of each year. In which completed year does it first exceed 1000?
3 marks
Solve ${(0.8)}^{t} < 0.3$ for real t.
3 marks
Can $2^{x} = - 3$ have a real solution? Explain.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Solve $8^{x} = 12$ to four decimal places.
2 marks
$x = log 12 / log 8$ .
x approximately 1.1950
Solve $2^{x + 3} = 7$ to four decimal places.
2 marks
$x + 3 = log 7 / log 2$ , then subtract 3.
x approximately -0.1926
Solve $4 {(1.2)}^{t} = 10$ to three decimal places.
2 marks
Isolate ${(1.2)}^{t} = 2.5$ ; use $t = log 2.5 / log 1.2$ .
t approximately 5.026
A 600-unit quantity grows by 8% at the end of each year. In which completed year does it first exceed 1000?
3 marks
$t > log (1000 / 600) / log 1.08 \approx 6.637$ .
Year 7 is the first integer year; substitution verifies the threshold.
7
Solve ${(0.8)}^{t} < 0.3$ for real t.
3 marks
Take logs and divide by $log 0.8 < 0$ , reversing the inequality.
$t > log 0.3 / log 0.8 \approx 5.396$
Can $2^{x} = - 3$ have a real solution? Explain.
2 marks
$2^{x}$ is positive for every real x, so it cannot equal a negative number.
No.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: exponential and logarithmic functions, growth models and numerical/algebraic prerequisites.

Official curriculum source ↗

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