Question

Show that ${{10}^{\log x}}=x$?

Answer 1

In this question we have an exponential expression which has a function of logarithm as its power. We will consider the left-hand side of the expression as a variable $y$ and then take logarithm on both the sides and then use the property of logarithms and simplify it to get the required solution.

Complete step by step solution:
We have to show that:
${{10}^{\log x}}=x\to \left( 1 \right)$
Consider the left-hand side of the expression as $y$ therefore, we get:
$\Rightarrow y={{10}^{\log x}}\to \left( 2 \right)$
On taking log on both the sides of the expression, we get:
$\Rightarrow \log y=\log \left( {{10}^{\log x}} \right)$
Now we know the property of log that $\log {{a}^{b}}=b\log a$ therefore, on using this property, we get:
$\Rightarrow \log y=\log x\log 10$
Now we know the value of $\log 10=1$ therefore, on substituting, we get:
$\Rightarrow \log y=\log x\times 1$
On simplifying, we get:
$\Rightarrow \log y=\log x$
Now since there is log on both the sides, we can take the antilog and write it as:
$\Rightarrow y=x\to \left( 3 \right)$
From equation $\left( 2 \right)$, we know that $y={{10}^{\log x}}$ therefore on substituting it in equation $\left( 3 \right)$, we get:
${{10}^{\log x}}=x$, which is the required solution, hence proved.

Note: It is to be noted that the logarithm we are using has the base $10$, the base is the number to which the log value has to be raised to, to get the original term. This is also called the antilog of the number which is the logical reverse of taking a log.
The most commonly used bases in logarithm are $10$ and $e$ which has a value of approximate $2.713...$
Logarithm to the base $e$ is also called as the natural log and also written as $\ln$.
Logarithm is used to simplify a mathematical expression, it converts multiplication to addition, division to subtraction and exponents to multiplication.