Question

How do you solve the exponential equation ${{4}^{2x}}={{4}^{x+2}}$ ?

Answer 1

Hint: Since the bases are the same in the given exponential equation on both sides (L.H.S. and R.H.S.), we can drop the bases and set the exponents equal to each other, i.e.
$\begin{align}
  & {{B}^{M}}={{B}^{N}} \\
 & \Rightarrow M=N \\
\end{align}$
Where, B=base, M=exponent_1 and N=exponent_2,

Complete step by step answer:
Following is the method one needs to follow to solve the given exponential equation-
Firstly we need to determine whether both L.H.S. and R.H.S. are written with the same base. In this case 4 can be written using the base 2 on both sides. Thus, we can rewrite the given equation in question as
$\Rightarrow {{\left( {{2}^{2}} \right)}^{2x}}={{\left( {{2}^{2}} \right)}^{x+2}}......(i)$
 Now we will use the properties of exponents to simplify the given exponential equation. Property that we need to use here is when a power is raised to a power, we multiply both the powers in equation (i).
$\Rightarrow {{2}^{4x}}={{2}^{2x+4}}.....(ii)$
Since the bases are the same on both the sides, we can drop the bases and set the exponents on both sides equal to each other. We will get,
$\Rightarrow 4x=2x+4$
Now we will solve the above linear equation to get the value of ‘x’.
First, move ‘$2x$’ to L.H.S. and subtract it from ‘$4x$ ‘.
$\Rightarrow 2x=4$
Then, divide 4 written in R.H.S. by 2 to get the value of ‘x’.
 $\Rightarrow x=\dfrac{4}{2}$
$\therefore x=2$
Hence, the solution of ${{4}^{2x}}={{4}^{x+2}}$ is $x=2$ .

Note:
Alternatively, we can use common logarithms on both sides, i.e.
$\log {{\left( 4 \right)}^{2x}}=\log {{\left( 4 \right)}^{x+2}}$
Using property ${{\log }_{a}}{{x}^{y}}=y{{\log }_{a}}x$ ,
$\Rightarrow \left( 2x \right)\log 4=\left( x+2 \right)\log 4$
Dividing each side by log4,
$\Rightarrow 2x=x+2$
We get,
$\Rightarrow x=2$