Question

Express the following in logarithmic form: $ 81 = {3^4} $
A. $ {\log _3}81 = 4 $
B. $ {\log _2}81 = 9 $
C. $ 2{\log _3}9 = 4 $
D. $ 4{\log _9}3 = 2 $

Answer 1

Hint: When we have to express any equation in form of logarithm then by using the formula of $ y = {a^x} $ we will get the value in form of $ {\log _a}y = x $ . Remember that logarithmic expressions are inverse of exponential expressions and vice versa.

Complete step-by-step answer:
Given, the expression is: $ 81 = {3^4} $
(A) As we know that when
 $ y = {a^x} $ Then,
 $\Rightarrow {\log _a}y = x $ …………..(i)
If we compare the given expression with the formula then we get,
 $ y = 81,x = 4 $ and $ a = 3 $
So substituting the value in equation (1) then we will get,
 $\Rightarrow {\log _3}81 = 4 $
Hence option A is correct.

(B) Now, if we take the expression as: $ 81 = {3^4} = {\left( {{3^2}} \right)^2} = {9^2} $ .
Then we will take the expression to express in logistic form.
 $ 81 = {9^2} $
Converting the equation in logistic form we will get,
 $\Rightarrow {\log _9}81 = 2 $
Hence, Option B is also correct.

(C) Now, again we take the expression as: $ 81 = {3^4} $ .
Then we will take the expression to express in logistic form.
 $
\Rightarrow 81 = {9^2}\\
{9^2} = {3^4}
$
Converting the equation in logistic form we will get,
$\Rightarrow {\log _3}{9^2} = 4 $
When any value having exponent under the log function then the power multiplied in the function in this way: $ \log {m^n} = n\log m $
So the expression becomes: $ 2{\log _3}9 = 4 $
Hence, Option C is also correct.

(D) Now, again we take the expression as: $ 81 = {3^4} = {\left( {{3^2}} \right)^2} = {9^2} $ .
Then we will take the expression to express in logistic form.
 $
\Rightarrow 81 = {9^2}\\
{3^4} = {9^2}
$
Converting the equation in logistic form we will get,
 $\Rightarrow {\log _9}{3^4} = 2 $
When any value having exponent under the log function then the power multiplied in the function in this way: $ \log {m^n} = n\log m $
So the expression becomes: $ 4{\log _9}3 = 2 $
Hence, Option D is also correct.

So, the correct answer is “ALL THE FOUR OPTIONS”.

Note: When the function or equation is given in the form exponent then can be easily expressed in a logistic function by using $ {\log _a}y = x $ ,which is expressing that value y under the log function having base a then its value is x.