Question

The value of ${\log _3}27$ is equal to
$(a){\text{ 3}} \\\ (b){\text{ 9}} \\\ (c){\text{ 16}} \\\ (d){\text{ 25}} \\\$

Answer 1

Hint: In this question we have to find the value of the given logarithmic expression. Use the property of logarithm ${\log _a}b = \dfrac{{\log b}}{{\log a}}$ along with other basic properties of logarithm to get the answer.

Complete step-by-step answer:

Given equation is
${\log _3}27$
As we know ${\log _a}b = \dfrac{{\log b}}{{\log a}}$ so use this logarithmic property in above equation we have,
$\Rightarrow {\log _3}27 = \dfrac{{\log 27}}{{\log 3}} = \dfrac{{\log {3^3}}}{{\log 3}}$
Now we also know that $\log {a^b} = b\log a$ so use this logarithmic property in above equation we have,
$\Rightarrow {\log _3}27 = \dfrac{{\log {3^3}}}{{\log 3}} = \dfrac{{3\log 3}}{{\log 3}}$
Now cancel out log3 from the numerator and denominator we have.
$\Rightarrow {\log _3}27 = \dfrac{{3\log 3}}{{\log 3}} = 3$
So this is the required answer.
Hence option (A) is correct.

Note: Whenever we face such types of problems the key concept is to have a good gist of the logarithmic identities, some of them have been mentioned above. This concept will help you get on the right track to reach the answer.