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.