Year 12: Exponentials and logarithms

Exponential and logarithmic questions are mostly about structure. If you can see when to rewrite bases, when to apply log laws and when to isolate the exponential part first, the algebra becomes much calmer.

Keep equivalent forms in mind: exponential form, logarithmic form and index notation are just three views of the same relationship.

What to master

Evaluating logarithms exactly when bases are compatible.
Using log laws in the correct direction to simplify or condense expressions.
Solving exponential equations by rewriting both sides with a common base where possible.
Using logarithms only after you have isolated the exponential term.

Exam habit: once you solve a logarithmic equation, check the domain conditions from the original logs. A value that makes an argument non-positive is not valid.

Exponentials and logarithms tutorials

Revision focus

Exponential and logarithmic questions become much easier when you classify the task before starting. Ask whether you are evaluating a logarithm exactly, simplifying with log laws, solving an exponential equation, solving a logarithmic equation, or modelling growth and decay. Each of those tasks has a standard pattern, and confusion usually comes from mixing those patterns together rather than from the algebra itself.

Another high-value habit is to reverse-check the answer. If a logarithm equals a certain number, convert it back into exponential form and see whether the statement is true. If a model gives a time or population value, ask whether the rounding and units make sense in context. Domain restrictions, positive log arguments, and sensible growth factors are not side details in this topic. They are part of what makes the answer mathematically valid.

Skills to practise

One reason this topic can feel unstable is that the same relationship can be written in several different ways. For example, $2^{3} = 8$ , $\log_{2} (8) = 3$ and "8 is a power of 2" are all saying the same thing. Strong students move freely between these forms. That flexibility is what lets you spot when an exponential equation can be solved by rewriting bases, when log laws will simplify the working, and when a word problem needs a model before any solving can begin.

Worked example

Problem: Solve

5 x 2^{x} = 37

First isolate the exponential term: $2^{x} = 37 / 5 = 7.4$ .
This cannot be rewritten neatly as a power of 2, so logarithms are needed.
Take logs of both sides: $\log (2^{x}) = \log (7.4)$ .
Use the power rule: $x \log 2 = \log 7.4$ .
Solve for $x$ : $x = \log 7.4 / \log 2$ .
Using a calculator gives $x = 2.888 . . .$ .

Answer:

x = 2.89

to 3 decimal places.

Common mistakes

Taking logs too early when a common-base rewrite would have been faster.
Using log laws on sums, even though $\log (a + b)$ does not split into simpler logs.
Forgetting that logarithm arguments must stay positive in equations.
Rounding too early in modelling questions, especially when solving for time.

Study checklist

A strong revision habit is to classify each question before doing any algebra. Ask: is this an exact evaluation, a simplification, an exponential equation, a logarithmic equation, or a modelling question? That single decision usually determines the correct method. If you can choose the method calmly, the rest of the topic becomes much more predictable.

It also helps to read every final answer backward. If a logarithm result says the answer is 3, check that the corresponding base raised to 3 really gives the argument. If an exponential model predicts a time, decide whether the wording needs rounding or interpretation. Those reverse checks are fast and very reliable.