Question

How do you evaluate ${{\log }_{3}}\left( 27 \right)$?

Answer 1

Hint: First we write $27\;$ in exponential form. $27\;$should be written in such a way that it is in the powers of $3$. It is easier if we write it in powers of $3$so that we can cancel the logarithmic base with it. Use the laws of logarithms to remove the exponent and then simplify it.

Complete step by step solution:
The given logarithmic expression is,${{\log }_{3}}\left( 27 \right)$
Firstly, we write $27$in exponential form, as the power of $3$.
Since $27=3\times 3\times 3$
Now, on converting it into exponential form,
$\Rightarrow 27={{3}^{2}}\times 3$
Now put it back into a logarithmic expression.
$\Rightarrow {{\log }_{3}}\left( {{3}^{2}}\times 3 \right)$
Using the Law of the sum of logarithms,
 If we have a function,$f\left( x \right)={{\log }_{c}}\left( ab \right)$.Then we can write it as $f\left( x \right)={{\log }_{c}}a+{{\log }_{c}}b$.
Here,$a={{3}^{2}};b=3;c=3$
$\Rightarrow {{\log }_{3}}\left( {{3}^{2}}\times 3 \right)={{\log }_{3}}\left( {{3}^{2}} \right)+{{\log }_{3}}3$
Now, consider the first term.
By using the law of powers of logarithms,
If we have the function, $f\left( x \right)={{\log }_{a}}\left( {{b}^{c}} \right)$ .Then we can convert into power form as, $f\left( x \right)=c{{\log }_{a}}\left( b \right)$
Here $a=3;b=3;c=2$
On Substituting,
$\Rightarrow {{\log }_{3}}\left( {{3}^{2}} \right)=2{{\log }_{3}}3$
If the base and the logarithm value is the same, they cancel out to get $1$
$\Rightarrow {{\log }_{3}}3=1$
$\Rightarrow 2{{\log }_{3}}3=2$
Now, consider the second term,${{\log }_{3}}3$
Since the base and the logarithm value is the same, they cancel out to get $1$
$\Rightarrow {{\log }_{3}}3=1$
Now, putting it all together we get,
$\Rightarrow 2+1=3$
Hence, ${{\log }_{3}}\left( 27 \right)=3$

Note: The logarithm of a given constant $y$ is the exponent to which another fixed constant, the base $b$, must be raised, to produce that constant $y$.
$\Rightarrow {\log _b}({b^x}) = x$
One should ensure that the base of the given logarithms is the same before evaluating the expression using the laws of the logarithms. While using the subtraction law of logarithms ensure which term is written in the numerator and which in the denominator. Always the first term will be in the numerator whereas the second term in the denominator.