Question

How do you solve $2\log x=\log 25$?

Answer 1

Hint: First take the coefficient 2 inside the logarithm function on the left side of the equation. Then raise both the sides by the base power of log ‘e’ to eliminate the logarithm part. Do the necessary calculations to get the value of ‘x’.

Complete step by step solution:
For solving $2\log x=\log 25$, we have to find the value of ‘x’ for which it will satisfy the equation.
We know, ‘log’ of some value ‘m’ with the coefficient ‘n’ i.e. $n\log m$ can be simplified by taking ‘n’ as power of ‘m’ which can be written as, $n\log m=\log {{m}^{n}}$
Hence, considering our equation $2\log x=\log 25$,
$2\log x$ can be written as $2\log x=\log {{x}^{2}}$
Now, our equation becomes
$\Rightarrow \log {{x}^{2}}=\log 25$
Again we know, if ‘log’ of some value ‘m’ is equal to ‘log’ of some value ‘n’ i.e. $\log m=\log n$, then it can be simplified by raising the power of both the sides by the base power ‘e’ which can be written as, ${{e}^{\log m}}={{e}^{\log n}}$. It can be further simplified as $m=n$(cancelling out ‘e’ and ‘log’ from both the sides as base of ‘log’ is also ‘e’)
Now, coming back to our equation,
Raising the power of both the sides by ‘e’, we get
$\Rightarrow {{e}^{\log {{x}^{2}}}}={{e}^{\log 25}}$
cancelling out ‘e’ and ‘log’ from both the sides, we get
$\begin{align}
  & \Rightarrow {{x}^{2}}=25 \\
 & \Rightarrow x=\sqrt{25} \\
 & \Rightarrow x=5 \\
\end{align}$
This is the required solution.

Note: We could obtain two values of ‘x’ as $x=\pm 5$. But as we know in $\log x$, ‘x’ can not be negative. So we have to discard the value $x=-5$. Hence, $x=5$ is the only solution of the given question.