Exponential functions and the number $e$, the laws of logarithms and the relationship between exponentials and logarithms, solving equations of the form $a^x = b$, and using logarithms to linearise data.

What this dot point is asking

CCEA wants you to understand exponential functions and the special number $e$, use the laws of logarithms, exploit the fact that logarithm and exponential are inverse operations to solve equations with the variable in the power, and use logarithms to linearise exponential or power-law data so a graph becomes a straight line. Exponential models of growth and decay run through both the pure and the applied content.

The answer

Exponential functions and the number e

The laws of logarithms

Solving equations of the form a^x = b

Because logarithm and exponential are inverse operations, taking logs of both sides of $a^x = b$ brings the power down: $x\log a = \log b$, so $x = \dfrac{\log b}{\log a}$. For equations involving $e$, use the natural logarithm.

Linearising data with logarithms

Worked example: finding constants from a log graph

Examples in context

Example 1. Compound interest. An investment growing at $5\%$ a year follows $A = P(1.05)^t$. To find when it doubles, set $2 = 1.05^t$ and take logs: $t = \log 2 / \log 1.05 \approx 14.2$ years. The logarithm is the only way to release a variable trapped in the exponent.

Example 2. Radioactive decay. A decaying mass $m = m_0 e^{-\lambda t}$ has a constant half-life found from $\tfrac{1}{2} = e^{-\lambda t_{1/2}}$, giving $t_{1/2} = \ln 2 / \lambda$. The same exponential-then-log technique appears in physics, finance and biology.

Try this

Q1. Write $\log 8 + \log 5$ as a single logarithm. [1 mark]

• Cue. $\log 40$.

Q2. Solve $2^x = 50$, to three significant figures. [3 marks]

• Cue. $x = \log 50 / \log 2 \approx 5.64$.

Q3. Simplify $\ln(e^3)$. [1 mark]

• Cue. $3$.