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# How do you evaluate ${{\log }_{9}}729$?

Last updated date: 12th Aug 2024
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Hint: Evaluating logarithmic equations we have to first check for the logarithm’s base. In this case we have the logarithm base as 9, so we will express the number given as a power of the base of the logarithm. Then using the logarithm formula, we will solve the equation.

Complete step-by-step solution:
Logarithm equation can be explained as an equation consisting of logarithm in the expression. To solve this expression, we will use the logarithm formula which is as follows:
Suppose we have, ${{a}^{x}}=b$, then using logarithm to write this expression we have,
${{\log }_{a}}b=x$
Here, we have ‘a’ as the logarithm base. We can say that, ‘x’ is the logarithm of ‘b’ to the base ‘a’.
Let us understand this property using an example.
For example – we know that the product of two written thrice is eight. It can be written as:
${{2}^{3}}=8$
If we use logarithm in the above expression, we can rewrite it as:
${{\log }_{2}}8=3$
So we have the base of the logarithm as 2, and since 8 can be written as two raised to the power 3. We get the following answer as 3.
$\Rightarrow {{\log }_{2}}{{2}^{3}}=3{{\log }_{2}}2=3$, as ${{\log }_{2}}2=1$
According to the question we have, we have to solve for ${{\log }_{9}}729$,
We can see here that the base of the logarithm is 9, so we will write the number 729 as a power of 9. It can be written as:
$729=9\times 9\times 9={{9}^{3}}$
Therefore, we have
${{\log }_{9}}729$
$\Rightarrow {{\log }_{9}}{{9}^{3}}=3{{\log }_{9}}9=3$
Therefore, ${{\log }_{9}}729=3$
We can confirm this as ${{\log }_{9}}729=3$ would mean, ${{9}^{3}}=729$ and which is correct.

Note: We can also solve the equation by taking,
${{\log }_{9}}729=y$
As we know that, ${{\log }_{a}}b=x$ can also be written as ${{a}^{x}}=b$. So, we have
${{9}^{y}}=729$
Writing the above expression as a power of same bases, we can have
$\Rightarrow {{({{3}^{2}})}^{y}}={{(3)}^{6}}$
$\Rightarrow {{(3)}^{2y}}={{(3)}^{6}}$
Since, the bases are same, we can equate their powers, we get
$\Rightarrow 2y=6$
$\Rightarrow y=3$
Therefore, ${{\log }_{9}}729=3$