Introduction
When an unknown appears in an exponent (index), ordinary algebraic operations cannot isolate it directly. Logarithms provide the bridge: by applying a logarithm to both sides of an equation or inequality, the index "comes down" as a factor via the power rule of logarithms. This is one of the most frequently examined techniques in Pure Mathematics 3, appearing in a wide variety of forms — from a simple single-base equation such as , to more complex structures like , to strict or non-strict inequalities.
Mastery of this topic relies directly on confident use of the laws of logarithms, which are assumed known.
Core Concept
The key principle is the logarithmic power rule:
When both sides of an equation are positive exponentials, take logarithms of both sides (base 10 or base — both work; is often neater) and apply the power rule to bring each index down as a coefficient. You then solve a linear equation in the unknown.
For inequalities, the same process applies, but you must be careful about the direction of the inequality sign:
- Taking of both sides preserves the inequality direction (since is an increasing function and its argument is positive).
- No direction reversal is needed for standard exponential inequalities.
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