Solve exponential equations by taking logarithms, and solve logarithmic equations using the definition and laws of logarithms

What this dot point is asking

SEAB wants you to solve two related kinds of equation: exponential equations, where the unknown sits in the exponent (like $3^x = 20$), and logarithmic equations, where the unknown sits inside a logarithm (like $\log_2(x + 1) = 3$). The first is solved by taking logs to bring the exponent down; the second by converting to index form or combining logs first. The unifying idea is that logs and powers are inverse operations, so each undoes the other.

The answer

Exponential equations with matching bases

If both sides can be written to the same base, equate the indices directly. For $2^x = 16$, write $16 = 2^4$, so $x = 4$. This is the quickest route whenever it is available.

Exponential equations needing logs

When the bases cannot be matched, take logarithms of both sides and use the power law to bring the exponent to the front:

Any base of logarithm works as long as you use it on both sides; the common (base 10) log on your calculator is the usual choice.

Logarithmic equations: convert to index form

A single logarithm equal to a number is solved by switching to index form using the definition $\log_a b = c \Leftrightarrow a^c = b$:

Logarithmic equations: combine first

If there are several log terms, use the laws to combine them into a single logarithm, then convert to index form. For example $\log x + \log(x - 3) = 1$ (base 10) becomes $\log\big(x(x - 3)\big) = 1$, so $x(x - 3) = 10$, a quadratic to solve.

Always check for invalid solutions

You can only take the log of a positive number, so after solving a logarithmic equation, reject any solution that makes the inside of a log zero or negative. This final check is often worth a mark.

Examples in context

Example 1. Population doubling. A population modelled by $P = 500(2)^t$ reaches $4000$ when $2^t = 8$, so $t = 3$ time units. When the target does not give a neat power, you take logs instead, which is exactly the exponential-equation method applied to a real growth problem.

Example 2. Radioactive decay. A decaying quantity $A = A_0(0.5)^t$ falls to a given level when you solve for $t$ by taking logs. Decay and growth questions are the most common real settings for these equations, tying the algebra to science contexts.

Try this

Q1. Solve $2^x = 64$. [1 mark]

• Cue. $64 = 2^6$, so $x = 6$.

Q2. Solve $5^x = 12$, to 3 significant figures. [2 marks]

• Cue. $x = \dfrac{\log 12}{\log 5} = 1.54$.

Q3. Solve $\log_4(2x - 1) = 2$. [2 marks]

• Cue. $2x - 1 = 4^2 = 16$, so $2x = 17$ and $x = 8.5$.