Question

evaluate log base root 2 32

Answer 1

The value of log base root 2 32 is 5/2

Answer 2

$\displaystyle \sf{ log_{ \sqrt{2} }(32) } = 10$

Given :

$\displaystyle \sf{ log_{ \sqrt{2} }(32) }$

To find :

The value of the expression

Formula :

We are aware of the formula on logarithm that

$\sf{1. \: \: \: log( {a}^{n} ) = n log(a) }$

$\sf{2. \: \: log_{a}(a) = 1}$

Solution :

Step 1 of 2 :

Write down the given expression

The given expression is

$\displaystyle \sf{ log_{ \sqrt{2} }(32) }$

Step 2 of 2 :

Simplify the given expression

$\displaystyle \sf{ log_{ \sqrt{2} }(32) }$

$\displaystyle \sf{ = log_{ \sqrt{2} } ( {2}^{5} ) }$

$\displaystyle \sf{ = log_{ \sqrt{2} } \: \bigg[ { \bigg({( \sqrt{2} ) }^{2} \bigg) }^{5} \bigg] }$

$\displaystyle \sf{ = log_{ \sqrt{2} } \: \bigg[ { \bigg( \sqrt{2} \bigg) }^{(2 \times 5)} \bigg] }$

$\displaystyle \sf{ = log_{ \sqrt{2} } \: { \bigg( \sqrt{2} \bigg) }^{10} }$

$\displaystyle \sf{ = 10 \: log_{ \sqrt{2} } \: ( \sqrt{2} )\: \: \: \bigg[ \: \because \: log( {a}^{n} ) = n log(a)\bigg] }$

$\displaystyle \sf{ = 10 \times 1\: \: \: \bigg[ \: \because \: log_{a}(a) = 1\bigg] }$

$= 10$

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