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How do you evaluate ${{\log }_{14}}\left( 14 \right)$?

Last updated date: 29th Feb 2024
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IVSAT 2024
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Hint: For evaluating the expression given in the above question, we need to write it in the form of an equation. For this we can let it equal to some variable, say \[y\], so that we can write the equation \[y={{\log }_{14}}\left( 14 \right)\]. Then we have to take the anti logarithm on both the sides. The antilogarithm function is equivalent to raising the given base of the logarithm function to the argument. Then on comparing both sides of the equation obtained, we can determine the required value of the expression.

Complete step by step answer:
Let us write the expression given in the question as
$\Rightarrow y={{\log }_{14}}\left( 14 \right)$
Now, the base of the given logarithm function is equal to $14$. So on taking the anti logarithm on both sides of the above equation to get
$\Rightarrow {{14}^{y}}=14$
We know that a number raised to the power of one is equal to the number itself. Therefore we can write $14={{14}^{1}}$ on the RHS of the above equation to get
$\Rightarrow {{14}^{y}}={{14}^{1}}$
Since the bases on both sides are equal to $14$, therefore on comparing the exponents of both sides of the above equation we finally get
$\Rightarrow y=1$

Hence, the given expression ${{\log }_{14}}\left( 14 \right)$ is equal to $1$.

Note: Do not forget to check the value of the base which is mentioned as the subscript to the logarithm function. Generally, when there is no base mentioned, we take it to be equal to $10$. But in this case it is mentioned to be equal to $14$. Also, we can also use the property of the logarithm to directly evaluate the given expression, which is given as ${{\log }_{a}}\left( a \right)=1$.
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