Question

How do you convert $ {\log _b}216 = 3 $ into the exponential form?

Answer 1

Hint: In this question, we are given a logarithm function, the logarithm functions are of the form $ y = {\log _x}a $ , we have to convert this function into an exponential function, the exponential functions are the inverse of the logarithm functions, they are of the form $ a = {x^y} $ ,that is, in exponential function, one term is raised to the power of another term. So, by using the laws of the logarithm, we have to convert this function into the exponential function. Then on comparing the left-hand side and right-hand side, we can also find the value of b. Using the above-mentioned concept, we can solve this question.

Complete step-by-step answer:
We know that –
 $
  if,\,{\log _n}x = a \\
   \Rightarrow x = {n^a} \;
  $
So,
 $
  {\log _b}216 = 3 \\
   \Rightarrow 216 = {(b)^3} \;
  $
We can find the value of b as follows –
 $ 216 = 6 \times 6 \times 6 = {6^3} $
On comparing the left-hand side and the right-hand side, we get –
 $ b = 6 $
Hence, the exponential form of $ {\log _b}216 = 3 $ is $ 216 = {(b)^3}\,or\,216 = {6^3} $
So, the correct answer is “ $ 216 = {(b)^3}\,or\,216 = {6^3} $ ”.

Note: The logarithm functions obey certain laws that are called laws of the logarithm; we can write the function in a variety of ways by using these laws. There are three laws of the logarithm, two of the laws are for addition and subtraction of two or more logarithm functions and the third law is to convert logarithm functions to exponential functions. The base of the given function is b, which is unknown so to express it as a proper exponential function; we can find the value of b by prime factorization of 216 and then write it in exponent form such that its power is 3.