Question

How do you evaluate ${{\log }_{9}}(3)$?

Answer 1

Hint: In this question, we have to simplify a logarithm function. Thus, we will apply the logarithm formula to get the required solution. So, first we will use the logarithm formula ${{\log }_{b}}a=\dfrac{\log a}{\log b}$ in the equation. As we know, 9 is the square of 3, thus we will apply it in the denominator. After that, we again apply the logarithm formula $\log {{a}^{b}}=b\log a$ in the equation and then we will make the necessary calculations, which is the required answer.

Complete step by step answer:
According to the question, we have to simplify a logarithmic function.
So, to solve this problem, we will use the logarithm formula.
The function given to us is ${{\log }_{9}}(3)$ ---------- (1)
Now, we will apply the logarithm formula ${{\log }_{b}}a=\dfrac{\log a}{\log b}$ in equation (1), we get
$\Rightarrow {{\log }_{9}}(3)=\dfrac{\log 3}{\log 9}$
As we know that 9 is the square of 3, thus we will apply the same rule in the denominator, thus we get
$\Rightarrow \dfrac{\log 3}{\log {{3}^{2}}}$
Now, we will apply the logarithm formula $\log {{a}^{b}}=b\log a$ in the above equation, we get
$\Rightarrow \dfrac{\log 3}{2.(\log 3)}$
As we know, in the division, the same terms will cancel out and we get the remainder 0, therefore we get
$\Rightarrow \dfrac{1}{2}$

Therefore, for the function ${{\log }_{9}}(3)$, we get the value $\dfrac{1}{2}$ , which is our required answer.

Note: While solving this problem, mention all the steps and formulas you are using while solving your problem. One of the alternative methods to solve this problem is converting the logarithm into the exponential function.
An alternative method:
The function: ${{\log }_{9}}(3)$
Let us first rewrite the above function as an equation, we get
${{\log }_{9}}(3)=x$ ---------- (1)
So, we will rewrite the equation (1) into an exponential equation because if ${{\log }_{a}}b=x$ then ${{a}^{x}}=b$ , therefore we get
$\Rightarrow {{9}^{x}}=3$
Since we can write 9 as the square of 3, we get
$\Rightarrow {{\left( {{3}^{2}} \right)}^{x}}=3$
Now, we will apply the exponent rule ${{\left( {{a}^{b}} \right)}^{c}}={{a}^{bc}}$ in the above equation, we get
$\Rightarrow {{3}^{2x}}=3$
Since the bases are the same in the above equation, the two terms are equal if their powers will be equal, thus we get
$\Rightarrow 2x=1$
Now, we will divide 2 on both sides in the above equation, we get
$\Rightarrow \dfrac{2}{2}x=\dfrac{1}{2}$
Thus, on further simplification, we get
$\Rightarrow x=\dfrac{1}{2}$ which is our required solution.