Show that the logarithmic function f:R0^+ R given by f(x) = _a x,… - 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
$gof(x) = gof(y)$
$\Rightarrow g(f(x)) = g(f(y))$
$\Rightarrow f(x) = f(y) [\because g$ in one-one$]$
$\Rightarrow x = y [\because f$ in one-one$]$
$\therefore$ gof is one-one function.

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