Question

How do you evaluate ${{\log }_{625}}25$?

Answer 1

Hint: Logarithms are easier to understand if we think about an expression giving in log form as asking a question. In ${{\log }_{625}}25$, the question being asked is; “What power/index of $625$ will give $25$? Or how do I make $625$ into $25$?”. We should recognise $25$ as being the square root of $625$.

Complete step by step answer:
For answering this question we will write the given value $25$ in terms of $625$ because the base here is $625$ .
A square root can be written as an index, $\sqrt{x}={{x}^{\dfrac{1}{2}}}$
By using the above formula we can get the answer.
$\sqrt{625}={{625}^{\dfrac{1}{2}}}$
By simplifying this we will have $25={{625}^{\dfrac{1}{2}}}$
Now, the given question will be, ${{\log }_{625}}25={{\log }_{625}}{{625}^{\dfrac{1}{2}}}$
$\Rightarrow {{\log }_{625}}25=\dfrac{1}{2}$

Therefore we can conclude that ${{\log }_{625}}25=\dfrac{1}{2}$.

Note: We should be well aware of the logarithmic properties. We should also be well aware of which logarithmic property to be used to the given question. We should be very careful while doing the calculation part of logarithms. The alternative method for answering this question is by interchanging the log form and index form using the concept ${{\log }_{a}}b=c\text{ }\Leftrightarrow \text{ }{{\text{a}}^{c}}=b$. By using the above formula we can solve the given question. Let us assume that ${{\log }_{625}}25=x$. Now by using the above mentioned formula we get the below equation ${{\log }_{625}}25=x\text{ }\Leftrightarrow \text{ 62}{{\text{5}}^{x}}=25$ .We can write ${{625}^{x}}$ as ${{\left( {{25}^{2}} \right)}^{x}}$.By equating the powers or indices we get the below equation, $2x=1$ $\Rightarrow x=\dfrac{1}{2}$. In both the methods we got the same value, ${{\log }_{625}}25=\dfrac{1}{2}$.