# 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}} \\

$

Last updated date: 28th Mar 2023

•

Total views: 306k

•

Views today: 5.82k

Answer

Verified

306k+ views

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 $a = {e^b}$.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE