Question

Find the value of ${\log _5}125$.

Answer 1

Hint- Here we will proceed by equating the value with x. Then we will use logarithmic property of a product to solve this question i.e. $\ln \left( {{x^y}} \right) = y \times \ln \left( x \right)$ . Hence we get the desired result.

Complete step-by-step answer:
Power rule of logarithmic function-
The natural log of x raised to the power of y is times the ln of x.
$\ln \left( {{x^y}} \right) = y \times \ln \left( x \right)$
Let the expression be equal to x,
$x = {\log _5}\left( {125} \right)$
Now we will apply logarithmic property to find the value of the expression,
 i.e. ${\log _b}\left( x \right) = y$
$ \Rightarrow {b^y} = x$
Therefore, ${5^x} = 125$
Or ${5^x} = {5^3}$
Since the bases are the same, the two expressions are equal only if the exponents are also equal.
$ \Rightarrow x = 3$
Therefore, ${\log _5}125=3$.
Note- In order to solve this type of question, we must know the difference between common log and natural log. Also one can get confused with the rules of the natural log to be used so we must know all the rules and its properties.