Question

How do you calculate $\dfrac{{\log 25}}{{\log 5}}$ ?

Answer 1

Hint : Try to represent 25 in terms of 5 and then use logarithm properties to solve the question
Here, $\dfrac{{\log 25}}{{\log 5}}$ must be first rewritten by changing 25 and writing it in the terms of 5 to modify the question and make it simple to understand. After modifying the given question, we will apply the property of log that is ${\log _y}{x^z} = z{\log _y}x$ on the modified version of the question to take terms out of it. After which we will cancel out the like terms which will finally yield our answer.

Complete step-by-step answer :
Here, the given question is to calculate the value of $\dfrac{{\log 25}}{{\log 5}}$
We know that, $25 = {5^2}$
Therefore, rewriting 25 in the given fraction to modify it:-
$\dfrac{{\log 25}}{{\log 5}} = \dfrac{{\log {5^2}}}{{\log 5}}$
Now, using logarithm property that ${\log _y}{x^z} = z{\log _y}x$ in the above form , we will get:-
$\dfrac{{\log {5^2}}}{{\log 5}} = \dfrac{{2\log 5}}{{\log 5}} \\\ = 2 \times \dfrac{{\log 5}}{{\log 5}} \\\$
Cancelling out the like terms from the above form, we will get
$2 \times \dfrac{{\log 5}}{{\log 5}} = 2$
Hence, the value of $\dfrac{{\log 25}}{{\log 5}}$ is 2.
So, the correct answer is “2”.

Note : Here, the given question is to calculate the value of $\dfrac{{\log 25}}{{\log 5}}$
We know that, $5 = \sqrt {25} = {25^{\dfrac{1}{2}}}$
Therefore, rewriting 5 in the given fraction to modify it:-
$\dfrac{{\log 25}}{{\log 5}} = \dfrac{{\log 25}}{{\log {{25}^{\dfrac{1}{2}}}}}$
Now, using logarithm property that ${\log _y}{x^z} = z{\log _y}x$ in the above form , we will get:-
$\dfrac{{\log 25}}{{\log {{25}^{\dfrac{1}{2}}}}} = \dfrac{{\log 25}}{{\dfrac{1}{2}\log 25}} \\\ = \dfrac{1}{{\dfrac{1}{2}}} \times \dfrac{{\log 25}}{{\log 25}} \\\ = 2 \times \dfrac{{\log 25}}{{\log 25}} \\\$
Cancelling out the like terms from the above form, we will get
$2 \times \dfrac{{\log 5}}{{\log 5}} = 2$
Hence, the value of $\dfrac{{\log 25}}{{\log 5}}$ is 2.