Question

If ${{4}^{x}}\text{ }-\text{ }{{\text{4}}^{x-1}}\text{ = 24}$, then $x$ is $\dfrac{\text{m}}{2}$. Find the value of m.

Answer 1

Hint: Reduce the given equation by taking ${{4}^{x}}$common. Then, to bring x to the base from the power, take logarithm. After finding the value of x, equate it with$\dfrac{\text{m}}{2}$.

Complete step-by-step answer:
The given equation is,
${{4}^{x}}\text{ }-\text{ }{{\text{4}}^{x-1}}\text{ = 24 }.....\text{(i)}$
Taking ${{4}^{x}}$from the left hand side of the equation, we get,
$\begin{align}
  & {{4}^{x}}(1\text{ }-\text{ }{{\text{4}}^{-1}})\text{ = 24} \\
 & \Rightarrow \text{ }{{4}^{x}}(1\text{ }-\text{ }\dfrac{1}{4})\text{ = 24} \\
 & \Rightarrow \text{ }{{4}^{x}}\text{ x }\dfrac{3}{4}\text{ = 24} \\
 & \Rightarrow \text{ }{{\text{4}}^{x}}\text{ = 32} \\
 & \Rightarrow \text{ }{{\text{4}}^{x}}\text{ = }{{\text{4}}^{\dfrac{5}{2}}} \\
\end{align}$

Taking logarithm to the base 4 on both sides of the equation, we get,
$\begin{align}
  & {{\log }_{4}}{{4}^{x}}\text{ = lo}{{\text{g}}_{4}}{{4}^{\dfrac{5}{2}}} \\
 & \therefore \text{ }x\text{ = }\dfrac{5}{2} \\
\end{align}$

Hence, we obtain the value of x as $\dfrac{5}{2}$. Now, we know the value of x is equal to $\dfrac{\text{m}}{2}$.
Thus, equating the two values, we get,
$\begin{align}
  & \dfrac{\text{m}}{2}\text{ = }\dfrac{5}{2} \\
 & \therefore \text{ m = 5} \\
\end{align}$

Hence, the correct value of m is 5.

Note: Logarithm is a mathematical tool or operation, which helps us to bring the values in the power of a number to the base and simplify any equation. The general convention of taking logarithm bases are 10 or natural base e. But, in this given problem, since the power is on the number 4, it is more convenient to take 4 as the base of the logarithm. It is advisable to take logarithm on a product or division term, as then only an equation is simplified owing to the properties of logarithm. If logarithm is taken on a sum or difference term as in the equation (i), it would make calculations much more difficult.