Question

Find the value of $x$ if $\left[ {{4}^{2x}}+10 \right]\div 2=37$.

Answer 1

Hint: To find the value of $x$ we can use logarithm. Simplify the equation and take logarithm to get the final value of $x$ .

Complete step-by-step answer:
Simplifying the equation:
$\begin{align}
  & \left[ {{4}^{2x}}+10 \right]\div 2=37 \\
 & \left[ {{4}^{2x}}+10 \right]=74 \\
 & {{4}^{2x}}=64 \\
\end{align}$
Now, taking logarithm both sides we get,
$\log \left( {{4}^{2x}} \right)=\log \left( 64 \right)$
We can write
$64$ As ${{4}^{3}}$
Hence,
$\begin{align}
  & \log \left( {{4}^{2x}} \right)=\log \left( {{4}^{3}} \right) \\
 & 2x=3 \\
 & x=\dfrac{2}{3} \\
\end{align}$
Step2 : Verification.
We can verify by just putting the value of $x$ we got in the original equation.
$\begin{align}
  & \left[ {{4}^{2\times \dfrac{3}{2}}}+10 \right]\div 2=37 \\
 & \left[ 64+10 \right]\div 2=37 \\
 & 74\div 2=37 \\
 & 37=37 \\
\end{align}$
Hence, the given equation is proved. And we found the correct value of $x$.

Additional Information:
Logarithm is a powerful technique to solve very sophisticated problems. We have not assumed any specific value of the base of logarithm in this question. Answer is independent of the base of the logarithm. The value of x can be a whole number or a rational number depending upon the R.H.S.

Note: We can also do this question by using a hit and trial method. We can randomly choose some value of x. If L.H.S comes out to be large then we put a smaller value of x and continue this process until we get L.H.S equal to R.H.S. One thing to be noted here is we didn’t assume the base of logarithm in this question. The base of logarithm can be anything because we are dealing with the exponent which will remain the same in either case. Another thing to be noted here is that we can use any mathematical tool other than logarithm but the thing is logarithm makes the exponential equations simple and hence it is recommended to use them to deal with the exponential equations.