Question

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Answer 1

QUESTION:

$\displaystyle \sf \large log_{ \sqrt{a} }( \sqrt{ {a}^{ - \frac {8}{5} }} )$

ANSWER:

$\displaystyle \sf \large log_{ \sqrt{a} }( \sqrt{ {a}^{ - \frac {8}{5} }} ) = - \frac{8}{5}$

GIVEN AND TO FIND:

$\displaystyle \sf \large The \ value \ of \ log_{ \sqrt{a} }( \sqrt{ {a}^{ - \frac {8}{5} }} )$

EXPLANATION:

$\displaystyle \sf Let \ log_{ \sqrt{a} }( {a}^{ - \frac {8}{5} } ) = x$

$\pink{\boxed{ \large { \bold{log_{ a }(b ) = x \implies \ b = {a}^{x} }}}}$

$\leadsto\displaystyle \sf\sqrt{ {a}^{ - \frac {8}{5} }} = {( \sqrt{a}) }^{x}$

$\leadsto \displaystyle { \large{\sf {a}^{ - \frac {8}{5} \times \frac{1}{2} } = {( a}) }^{x \times \frac{1}{2} }}$

Bases are equal, so equate the powers.

$\leadsto\sf - \dfrac {8}{5} \times \dfrac{1}{2} = x \times \dfrac{1}{2}$

$\leadsto \sf x = - \dfrac {8}{5}$

$\bigstar\blue{\boxed{ \bold {\large log_{ \sqrt{a} }( \sqrt{ {a}^{ - \frac {8}{5} }} ) = - \frac{8}{5}}}}$