Worked example · Mathematics

Solving exponential equations with logarithms: 3^x = 20

Powers are easy to undo when the answer is exact: 3² = 9 and 3³ = 27. But what power of 3 gives 20? It lies somewhere between 2 and 3, and no amount of tidy factorising will name it. A logarithm brings the unknown exponent down where ordinary division can reach it.

GCSE to A-level
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Logarithms
≈ 7 minutes

Step 1 — on paper

When both sides can be written with the same base

Start with 3^x = 81. Since 81 = 3⁴, equal powers with the same positive base have equal exponents, so x = 4. The same idea solves 2^x = 1/8 because 1/8 = 2⁻³, giving x = −3.

3^{x} = 81 = 3^{4} \Rightarrow x = 4

Twenty is not an integer power of 3. We can still bracket the answer: 9 < 20 < 27, so 2 < x < 3. That estimate is the check the decimal answer must pass.

Step 2 — on the calculator

Take a logarithm of both sides

Logarithms turn powers into multiplication. Taking the natural logarithm of both sides does not change the equality, then the power rule moves x in front of the logarithm:

\begin{matrix} \ln (3^{x}) & = & \ln (20) \\ x \ln (3) & = & \ln (20) \\ x & = & \frac{\ln (20)}{\ln (3)} \end{matrix}

ln
20
)
÷
ln
3
)
=

ln(20)÷ln(3) 2.726833028 The power of 3 that produces 20

Ans on iPhone showing natural log 20 divided by natural log 3 and the result 2.726833028. — The entire numerator and denominator remain visible. Substituting the unrounded result back into 3^x returns 20.

Common log works too. log(20)÷log(3) gives the same result. The base cancels through the change-of-base formula; mixing bases between the top and bottom is the mistake. Use ln for both or log for both.

Step 3 — solve it by looking

The answer is where 3^x meets 20

Plot the two sides of the equation as separate functions. Their crossing is the x-value that makes them equal. Set the window to x from 0 to 4 and y from 0 to 30, then choose Intersection:

f₁(x) = 3^x f₂(x) = 20

Ans Graph showing the rising curve 3 to the power x crossing the horizontal line y equals 20 near x equals 2.73. — **Intersection** — 3^x reaches 20 at about **x = 2.73**, agreeing with ln(20)÷ln(3).

The curve also explains why there is only one real answer. For base 3, 3^x is positive and strictly increasing: every positive target height is crossed once. A target at zero or below would never be met.

Step 4 — make it a real question

When does a growing culture quadruple?

A model starts with 500 cells and grows by 18% each hour. Under the model, after t hours there are 500×1.18^t cells. To find when the count reaches 2,000:

500 \times {1.18}^{t} = 2000 \Rightarrow t = \frac{\ln (4)}{\ln (1.18)} = 8.375670267 h

The model reaches 2,000 after about 8 hours 23 minutes. The hourly checkpoints bracket it: after 8 hours the model gives 1,879 cells, and after 9 hours 2,218. If growth is only applied in whole hourly steps, the first recorded count at or above 2,000 is therefore the 9-hour reading; the fractional answer assumes growth acts continuously between those readings.

Checks for the two equations.
Question	Logarithm result	Substitution check
3^x = 20	x = 2.726833028	3^2.726833028 ≈ 20
500×1.18^t = 2000	t = 8.375670267 h	500×1.18^8.375670267 ≈ 2000
One hour earlier	t = 8	1,879.43 cells
One hour later	t = 9	2,217.73 cells

What the growth model assumes

The 18% fractional growth rate stays constant, with no shortage of nutrients, space or oxygen.
The population is treated as a continuous quantity even though cells are discrete.
The same exponential law is extended between hourly observations; a process updated only at whole hours gives a different operational answer.
The starting count and rate are treated as exact. Real biological measurements would limit the useful precision of the time.

Next: compound interest for the same mathematics attached to money, carbon dating for a decreasing exponential turned backwards with logarithms, or calculating pH for a logarithm used as a measurement scale.

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Worked on Ans — a scientific calculator you can own and Ans Graph. The logarithm gives the digits; the intersection shows why they are the answer. Both are paid for once.

Solving exponential equations with logarithms: 3x = 20