Logarithms: definition, properties and worked examples

A logarithm answers a single question: what exponent do I need to raise the base to in order to get this number?

with the existence conditions $b > 0$, $b \neq 1$ and $a > 0$.

First examples

• $\log_2 8 = 3$, because $2^3 = 8$.
• $\log_{10} 1000 = 3$, because $10^3 = 1000$.
• $\log_5 1 = 0$, because $5^0 = 1$.
• $\log_3 \tfrac{1}{9} = -2$, because $3^{-2} = \tfrac{1}{9}$.

Worked example: $\log_9 27$

1. Write both numbers as powers of 3: $9 = 3^2$ and $27 = 3^3$.
2. Look for $x$ with $9^x = 27$: $(3^2)^x = 3^3 \Rightarrow 3^{2x} = 3^3$.
3. $2x = 3$, so $\log_9 27 = \dfrac{3}{2}$.

Properties

For valid bases and arguments:

They all come from the rules of exponents: multiplying powers with the same base adds exponents, and a logarithm is exactly an exponent.

Worked example: $\log 12$ given $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$

1. Factor: $12 = 2^2 \cdot 3$.
2. $\log 12 = 2 \log 2 + \log 3$.
3. $\log 12 \approx 0.602 + 0.477 = 1.079$.

Worked example: $\log 5$ with the same data

1. $5 = \dfrac{10}{2}$.
2. $\log 5 = \log 10 - \log 2 \approx 1 - 0.301 = 0.699$.

Change of base

Calculators usually have only $\log$ (base 10) and $\ln$ (base $e$). For any other base, divide:

Worked example: $\log_2 3$

1. $\log_2 3 = \dfrac{\log 3}{\log 2}$.
2. $\approx \dfrac{0.477}{0.301} \approx 1.585$.
3. Rough check: $2^1 = 2 < 3 < 4 = 2^2$, so the value lies between 1 and 2.

Exponential equations

When you cannot match the bases, take the logarithm of both sides and use the power rule.

Worked example: $2^x = 10$

1. $\log 2^x = \log 10$.
2. $x \cdot \log 2 = 1$.
3. $x = \dfrac{1}{\log 2} \approx \dfrac{1}{0.301} \approx 3.32$.

Worked example: $3^{x+1} = 81$

1. $81 = 3^4$, so the bases can be matched: $x + 1 = 4$.
2. $x = 3$.

Common mistakes

1. Writing $\log(a + b) = \log a + \log b$.
2. Ignoring the existence conditions and accepting solutions that make the argument negative.
3. Confusing $\log_b x^n = n \log_b x$ with $(\log_b x)^n$.