Question

solve this logarithm ​

Answer 1

Given Question :-

Find the value of x, if

$\rm \: \dfrac{1}{ log_{x}(10) } + 2 = \dfrac{2}{ log_{5}(10) }$

$\large\underline{\sf{Solution-}}$

Given equation is

$\rm \: \dfrac{1}{ log_{x}(10) } + 2 = \dfrac{2}{ log_{5}(10) }$

We know,

$\boxed{\sf{ \: \: log_{a}(b) = \frac{1}{ log_{b}(a) } \: \: }}$

So, using this property, we get

$\rm \: log_{10}(x) + 2 = 2log_{10}(5)$

We know,

$\boxed{\sf{ \: \: log_{x}( {x}^{y} ) = y \: \: }}$

So, using this property, we get

$\rm \: log_{10}(x) + log_{10}( {10}^{2} ) = log_{10}( {5}^{2} )$

$\rm \: log_{10}(x) + log_{10}(100) = log_{10}(25)$

We know,

$\boxed{\sf{ \: log_{a}(m) + log_{a}(n) = log_{a}(mn) \: \: }}$

So, using this result, we get

$\rm \: log_{10}(100x) = log_{10}(25)$

$\rm\implies \:100x = 25$

$\rm\implies \:x = \dfrac{1}{4}$

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ADDITIONAL INFORMATION

$\boxed{\sf{ \: log_{a}(m) - log_{a}(n) = log_{a} \: \frac{m}{n} \: \: }}$

$\boxed{\sf{ \: log_{a}( {x}^{y} ) = y \: log_{a} \: x \: \: }}$

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

$\boxed{\sf{ \: log_{a}( {a}^{x} ) = x \: \: }}$

$\boxed{\sf{ \: log_{ {a}^{y} }( {a}^{x} ) = \frac{x}{y} \: \: }}$

$\boxed{\sf{ \: {a}^{ log_{a}(x) } \: = \: x \: \: }}$

$\boxed{\sf{ \: {a}^{y log_{a}(x) } \: = \: {x}^{y} \: \: }}$

Answer 2

Refer the Given Attachment.