Question

How do you solve $y=2\log \left( x \right)$?

Answer 1

Hint: First try to take the constant inside the logarithm on the right side of the equation. Then take $e$ to the power both the sides to eliminate the logarithm part. Do the necessary simplifications and solve for ‘x’ in terms of ‘y’.

Complete step by step answer:
Solving the equation means, we have to find the value of ‘x’ in terms of ‘y’, for which the equation gets satisfied.
Considering the equation $y=2\log \left( x \right)$
As we know $\log {{m}^{n}}=n\log m$ (by logarithmic rule)
So, $2\log \left( x \right)$ can be written as $2\log \left( x \right)=\log {{x}^{2}}$.
Now, our equation becomes
$\Rightarrow y=\log {{x}^{2}}$
Raising both the sides by the power of ‘$e$’, we get
$\Rightarrow {{e}^{y}}={{e}^{\log {{x}^{2}}}}$
Again, as we know ${{e}^{\log m}}=m$ (by logarithmic rule)
So, ${{e}^{\log {{x}^{2}}}}$ can be written as ${{e}^{\log {{x}^{2}}}}={{x}^{2}}$
Hence, the equation becomes
$\Rightarrow {{e}^{y}}={{x}^{2}}$
Which can also be written as
$\Rightarrow {{x}^{2}}={{e}^{y}}$
Taking square root both the sides, we get
$\begin{align}
  & \Rightarrow \sqrt{{{x}^{2}}}=\sqrt{{{e}^{y}}} \\
 & \Rightarrow x=\sqrt{{{e}^{y}}} \\
\end{align}$
So, we got in terms of ‘y’ as $x=\sqrt{{{e}^{y}}}$.

Note: As we know the base of log is also $e$, so while taking ${{e}^{\log m}}$ both ‘$e$’ and ${{\log }_{e}}$ in its power cancels out to give the result as ‘m’. Hence, ${{e}^{\log m}}=m$. Here we are getting the final value of ‘x’, in terms of ‘y’, because there are two unknown variables as ‘x’ and ‘y’ in the given equation and only one equation. So, to get both the values of ‘x’ and ‘y’ or the value of ‘x’ independent of ‘y’ and vice-versa, we need at least two equations for two different variables.