Question

The logarithm form of ${5^3} = 125$ is equal to
A. ${\log _5}125 = 3$
B. ${\log _5}125 = 5$
C. ${\log _3}125 = 5$
D. ${\log _5}3 = 3$

Answer 1

Hint : A logarithm is the inverse of function to exponentiation. This means that the logarithm of a given number x is the exponent to which another fixed number, base b, must be raised to produce that number x. If a power is in the form $a = {b^k}$ then the logarithm of a to the base b is equal to k, ${\log _b}a = k$ . Use this info to find the logarithm form of ${5^3} = 125$ .

Complete step by step solution:
Logarithm counts the no. of occurrences of the same factor in repeated multiplication, like 1000 is the product of three 10s which gives the no. of occurrences of 10 as 3 and this is our logarithm of 1000 with base 10. There are two types of logarithms; common logarithms and natural logarithms. Logarithms with base 10 are called common logarithms and logarithms with base ‘e’ are called natural logarithms.
We are given to find the logarithm form of ${5^3} = 125$
On comparing the given expression with $a = {b^k}$ , we get the value of a as 125, value of b as 5 and the value of k as 3.
Therefore, the logarithmic form of ${5^3} = 125$ is ${\log _b}a = k$ , this gives us ${\log _5}125 = 3$
So, the correct answer is “Option A”.

Note : In the value of ${\log _b}a$ , confirm that b is always greater than zero and never equal 1; a must be a positive real number. If ${\log _b}a = k$ , then $a = {b^k}$ and vice-versa. Do not confuse logarithm with an algorithm as they both almost sound similar. An algorithm is a procedure to produce a solution to a problem whereas a logarithm is an exponent.