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How do you solve for ${\log _x}125 = 3$?

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Last updated date: 01st Mar 2024
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IVSAT 2024
Answer
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Hint:This question is related to logarithm and related concepts. An exponent that is written in a special way is known as a logarithm. Logarithm functions are just opposite or inverse of exponential functions. We can easily express any exponential function in a logarithm form. Similarly, all the logarithm functions can be easily rewritten in exponential form. In order to solve this equation, we have to use some of the logarithm function properties.

Complete step by step solution:
Here, in this question we have to solve ${\log _x}125 = 3$ for the value of $x$.
This question deals with logarithm functions, which are just the inverse of exponential functions. In order to solve this question, we will have to make use of logarithm function properties and rules.
The power rule of logarithm function-
The natural log of $x$ raised to the power of $y$ is times the $\ln$ of $x$.
$\ln \left( {{x^y}} \right) = y \times \ln \left( x \right)$
Now, according to the question,
$ \Rightarrow 125 = {x^3}$------(1)
We know that $125$ can be written as $5 \times 5 \times 5$. So, $125 = {5^3}$.
Using the same in equation (1), we get,
$
\Rightarrow {5^3} = {x^3} \\
\Rightarrow 5 = x \\
$
Therefore, the value of $x$ is $5$.

Note: This problem and similar to these can very easily be solved by making use of different logarithm properties. Students should keep in mind the properties of logarithmic functions. Logarithms are useful when we want to work with large numbers. Logarithm has many uses in real life, such as in electronics, acoustics, earthquake analysis and population prediction. When the base of common logarithm is $10$ then, the base of a natural logarithm is number $e$.
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