Math · College algebra · Worked example

Solve an exponential equation with logarithms

Solve 3 · 5ˣ = 60, exactly and to three decimal places.

3 \cdot 5^{x} = 60

Isolate the power

Divide both sides by 3.

5^{x} = 20

5ˣ = 20 asks for the exponent that turns 5 into 20, which is log₅ 20 by definition.

x = {log}_{5} 20

Divide natural logarithms: ln 20 ≈ 2.9957 and ln 5 ≈ 1.6094.

x = \frac{ln 20}{ln 5} \approx 1.861

5^1.861 ≈ 20.0, and 3 × 20.0 = 60. Since 5² = 25 is more than 20, the answer should be a little less than 2, and it is.

x = log₅ 20 = ln 20/ln 5 ≈ 1.861.

x = {log}_{5} 20 \approx 1.861

At 5% continuous growth, how long does it take an amount to double? Solve e^(0.05t) = 2.

Show the answer and explanation

Take ln of both sides: 0.05t = ln 2, so t = ln 2/0.05 = 0.6931/0.05 ≈ 13.9 years.

\begin{array}{l} e^{0.05 t} = 2 \\ 0.05 t = ln 2 \\ t = \frac{ln 2}{0.05} \\ t \approx 13.9 \end{array}

Open the graph and find where y = 3 · 5ˣ meets the line y = 60: the intersection sits at x ≈ 1.861.

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