Exponential and logarithmic reasoning

This module covers how exponential and logarithmic ideas work together in Additional Mathematics: turning an index into a log and back, combining logs with the three laws, solving log equations while respecting the domain, straightening a curve with logs, and reading a growth constant correctly. One idea runs through all of it: a logarithm is an index, so every rule about logs is an index rule in disguise.

Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content, notation and calculator rules in your exam year. Our Additional Mathematics learning guide shows where this module sits among the others.

What should you already know?

You need the laws of indices, including negative and fractional powers, and confident algebra with quadratics. If quadratic work is slow, revise quadratic structure and discriminants first. The straight-line side of this module connects forward to straight lines and linearisation.

An orienting example

Solve log₃ x + log₃ (x − 2) = 1.

Step 1, domain: x > 0 and x − 2 > 0, so x > 2.

Step 2, combine: log₃ [x(x − 2)] = 1.

Step 3, convert to index form: x(x − 2) = 3¹ = 3.

Step 4, solve: x² − 2x − 3 = 0, so (x − 3)(x + 1) = 0 and x = 3 or x = −1.

Step 5, apply the domain: x = −1 is not greater than 2, so it is rejected.

Check: log₃ 3 + log₃ 1 = 1 + 0 = 1. ✓ The answer is x = 3.

That one question used the log laws, conversion to an index, a quadratic and a domain check. Each step matches a lesson below.

In which order should you study it?

1. Convert between exponential and logarithmic statements: the base idea, that a log is an index written from the other side.
2. Combine logarithms with correct coefficients: the three laws, so several logs can become one.
3. Solve an equation with a log-domain restriction: uses conversion and laws together, then rejects any root that breaks the domain.
4. Linearise an exponential relationship: turns y = abˣ or y = axⁿ into a straight line and reads the constants from it.
5. Distinguish a growth constant from percentage growth: separates k in e^kt from the percentage rise in one time period.

Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.

Which traps catch most students here?

• Swapping the base and the index when converting between forms.
• Multiplying inside a log when a coefficient should become a power.
• Splitting log (a + b) into two logs, which no law allows.
• Keeping an invalid root because the quadratic gave two answers.
• Copying the intercept as a on a log graph, when it is lg a.
• Reading k as the percentage growth instead of finding the yearly factor.

Each lesson shows one of these slips in full and then corrects it.

How should you use the practice set?

Attempt each question on paper first, and write the domain condition as its own line before you solve any log equation. Then open the worked answer and compare method as well as final value. Use the non-calculator working trainer to rehearse exact arithmetic, and the quadratic structure explorer to test the roots of a quadratic that appears in a log equation.

When you get something wrong, read the routing table at the end of the practice set and return to the lesson it names. Keep a record in the mistake log and retest queue, and retry a fresh question a few days later.

If your progress stalls on the same habit, a teacher can look at your written solutions in online one-to-one Additional Mathematics tuition.