Show that the logarithmic function f:R0^+ R given by f(x) = log_a… - Vidyadip

Question

Show that the logarithmic function $\text{f}:\text{R}0^+\rightarrow \text{R}$ given by $f(x) = log_a x, a > 0$ is a bijection.

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Answer

We have, $f : A \rightarrow B$ and $g : B \rightarrow C$ are one $-$ one functions.
Now we have to prove: gof : $A \rightarrow C$ in one $-$ one.
Let $\text{x, y}\in\text{A}$ such that
$\text{gof(x) = gof(y)}$
$\Rightarrow g(f(x)) = g(f(y))$
$\Rightarrow f(x) = f(y) \ [\because g$ in one $-$ one$]$
$\Rightarrow x = y \ [\because f $ in one $-$ one$]$
$\therefore \text{gof}$ is one $-$ one function.

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