Question

Determine the value of log 128 to the base √2

Answer 1

Let log 128 to the base √2 be x
Then,
$( \sqrt{2})^{x} = 128 \\\\ (2^{\frac{1}{2}})^{x} = 128 \\\\ (2^ \frac{x}{2} ) = 2^{7} \\\\ x/2 = 7 \\\\ x = 7(2) \\\\ x = 14 \\\\ log \:128 \:\:to \:\:the \:\:base \:\: \sqrt{2} = 14$

Answer 2

$\displaystyle \sf{ log_{ \sqrt{2} }(128) } = 14$

Given :

$\displaystyle \sf{ log_{ \sqrt{2} }(128) }$

To find :

Simplify the expression

Solution :

Step 1 of 2 :

Write down the given expression

The given expression is

$\displaystyle \sf{ log_{ \sqrt{2} }(128) }$

Step 2 of 2 :

Simplify the given expression

$\displaystyle \sf{ log_{ \sqrt{2} }(128) }$

$\displaystyle \sf{ = \frac{ log(128) }{ log( \sqrt{2} ) } }$

$\displaystyle \sf{ = \frac{ log( {2}^{7} ) }{ log( {2}^{ \frac{1}{2} } ) } }$

$\displaystyle \sf{ = \frac{ 7log( 2) }{ \frac{1}{2} log( {2}^{ } ) } }$

$\displaystyle \sf{ = \frac{ 7}{ \frac{1}{2} } }$

$\displaystyle \sf{ = 7 \times 2 }$

$\displaystyle \sf{ = 14 }$

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