Idea 2: applying an injective function

Given an equation $f(x) = g(x)$, we can apply to both sides an injective (that is, “invertible”) function without losing or gaining solutions. It is the idea behind many moves that during the year we have learnt “by heart”: squaring, cubing, removing exponentials or logarithms.

In brief — Examples of injective functions used during the year

• $t\mapsto t^2$ is injective on $[0,+\infty)$: if both sides are $\ge 0$, we can square while preserving the solutions.
• $t\mapsto t^3$ is injective on the whole of $\mathbb{R}$: we can always cube, without having to distinguish cases.
• $t\mapsto a^t$ (exponential) is injective on $\mathbb{R}$: if $a^{f(x)} = a^{g(x)}$, then $f(x) = g(x)$.
• $t\mapsto \log_a t$ (logarithm) is injective on $(0,+\infty)$: if $\log_a f(x) = \log_a g(x)$ and $f,g>0$, then $f(x) = g(x)$.

Remark

This idea explains at a glance why we can “remove” logarithms or exponential bases when they appear on both sides. It is the same idea, dressed up differently.

Links

Topics: Unifying methods
Concepts: Exponential equation · Irrational equation · Logarithmic equation · Injective function
Functions: Exponential function · Logarithmic function
Skills: Reasoning by cases · Solving equations