Question

log 2 with base 5 into log 5 with base 11 into log 11 with best 32 is equal to​

Answer 1

$\displaystyle \sf{ log_{5}(2) \times log_{11}(5) \times log_{32}(11)} = \frac{1}{5}$

Correct question :

$\displaystyle \sf{ log_{5}(2) \times log_{11}(5) \times log_{32}(11)}=$

Given :

$\displaystyle \sf{ log_{5}(2) \times log_{11}(5) \times log_{32}(11)}$

To find :

Simplify the given expression

Solution :

Step 1 of 2 :

Write down the given expression

Here the given expression is

$\displaystyle \sf{ log_{5}(2) \times log_{11}(5) \times log_{32}(11)}$

Step 2 of 2 :

Simplify the given expression

$\displaystyle \sf{ log_{5}(2) \times log_{11}(5) \times log_{32}(11)}$

$\displaystyle \sf = \frac{log \: 2}{log \: 5} \times \frac{log \: 5}{log \: 11} \times \frac{log \: 11}{log \: 32} \: \: \: \bigg[ \: \because \: log_{x}(y) = \frac{log \: y}{log \: x} \bigg]$

$\displaystyle \sf{ = \frac{log \: 2}{log \: 32} }$

$\displaystyle \sf{ = \frac{log \: 2}{log \: {2}^{5} } }$

$\displaystyle \sf{ = \frac{log \: 2}{5log \: {2}^{} } }\: \: \: \bigg[ \: \because \: log_{x}( {y}^{m} ) = mlog_{x}(y) \bigg]$

$\displaystyle \sf{ = \frac{1}{5} }$

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Answer 2

Answer:

$log_5(2)\times log_{11}(5)\times log_{32}(11)$ is equal to  $\bold{\frac{1}{5} }$ .

Step-by-step explanation:

The given expression is $log_5(2)\times log_{11}(5)\times log_{32}(11)$

Now apply  $log_x(y)=\frac{log y}{log x}$

$log_5(2)\times log_{11}(5)\times log_{32}(11)=\frac{log 2}{log 5} \times \frac{log 5}{log 11}\times \frac{log 11}{log 32}$

Now cancel the like terms

$log_5(2)\times log_{11}(5)\times log_{32}(11)=\frac{log 2}{log 32}$

$log_5(2)\times log_{11}(5)\times log_{32}(11)=\frac{log 2}{log 2^5}$

Now apply the logarithm formula $log_x(y^m)=mlog_x(y)$, we have

$log_5(2)\times log_{11}(5)\times log_{32}(11)= \frac{log 2}{5log 2}$

$log_5(2)\times log_{11}(5)\times log_{32}(11)=\frac{1}{5}$