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.