# If a logarithm function is given as $\dfrac{{\log 225}}{{\log 15}} = \log x$, then what is the value of x.

$

{\text{A}}{\text{. 400}} \\

{\text{B}}{\text{. 300}} \\

{\text{C}}{\text{. 200}} \\

{\text{D}}{\text{. 100}} \\

$

Answer

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Hint- Here, we will be using the basic formula of the logarithm function which is $\log \left( {{a^b}} \right) = b\left( {\log a} \right)$ along with the condition that if $\log a = b$ then in order to get the value of x.

Complete step-by-step answer:

Given, $\dfrac{{\log 225}}{{\log 15}} = \log x{\text{ }} \to {\text{(1)}}$

Since $a = {\left( {10} \right)^b}$, the square of number 15 is equal to 225 i.e., $225 = {\left( {15} \right)^2}$

Now, equation (1) becomes

$\dfrac{{\log \left[ {{{\left( {15} \right)}^2}} \right]}}{{\log 15}} = \log x{\text{ }} \to {\text{(2)}}$

As we know that $\log \left( {{a^b}} \right) = b\left( {\log a} \right)$

Using the above mentioned formula, equation (2) becomes

$

\dfrac{{2\log 15}}{{\log 15}} = \log x \\

\Rightarrow 2 = \log x \\

\Rightarrow \log x = 2{\text{ }} \to {\text{(3)}} \\

$

Also we know that if $\log a = b$, then $a = {\left( {10} \right)^b}$

Using the above formula, equation (3) becomes

$ \Rightarrow x = {\left( {10} \right)^2} = 100$

So, the required value of x is 100.

Hence, option D is correct.

Note- In this particular problem, we need to make sure that the given equation consists of the log function not ln function because both of these functions are different. For log function, the condition is that if $\log a = b$ then $a = {\left( {10} \right)^b}$ and for ln function, the condition is that if $\ln a = b$ then $a = {e^b}$.

Complete step-by-step answer:

Given, $\dfrac{{\log 225}}{{\log 15}} = \log x{\text{ }} \to {\text{(1)}}$

Since $a = {\left( {10} \right)^b}$, the square of number 15 is equal to 225 i.e., $225 = {\left( {15} \right)^2}$

Now, equation (1) becomes

$\dfrac{{\log \left[ {{{\left( {15} \right)}^2}} \right]}}{{\log 15}} = \log x{\text{ }} \to {\text{(2)}}$

As we know that $\log \left( {{a^b}} \right) = b\left( {\log a} \right)$

Using the above mentioned formula, equation (2) becomes

$

\dfrac{{2\log 15}}{{\log 15}} = \log x \\

\Rightarrow 2 = \log x \\

\Rightarrow \log x = 2{\text{ }} \to {\text{(3)}} \\

$

Also we know that if $\log a = b$, then $a = {\left( {10} \right)^b}$

Using the above formula, equation (3) becomes

$ \Rightarrow x = {\left( {10} \right)^2} = 100$

So, the required value of x is 100.

Hence, option D is correct.

Note- In this particular problem, we need to make sure that the given equation consists of the log function not ln function because both of these functions are different. For log function, the condition is that if $\log a = b$ then $a = {\left( {10} \right)^b}$ and for ln function, the condition is that if $\ln a = b$ then $a = {e^b}$.

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