Question

solve the exponential equation by expressing each side as a power of the same base and then solving for x.
$7^{\frac{x - 7}{4}} = \sqrt4{7}$
the solution set is \boxed{}

Explanation:

Step1: Rewrite the right - hand side

We know that $\sqrt[4]{7}=7^{\frac{1}{4}}$. So the equation $7^{\frac{x - 7}{4}}=\sqrt[4]{7}$ can be rewritten as $7^{\frac{x - 7}{4}} = 7^{\frac{1}{4}}$.

Step2: Set the exponents equal

Since the bases are the same (both are 7) and the exponential function $y = a^{x}$ ($a>0,a
eq1$) is one - to - one, if $a^{m}=a^{n}$, then $m = n$.
So we set $\frac{x - 7}{4}=\frac{1}{4}$.

Step3: Solve for x

Multiply both sides of the equation $\frac{x - 7}{4}=\frac{1}{4}$ by 4 to get $x-7 = 1$.
Then add 7 to both sides: $x=1 + 7=8$.

Answer:

The solution set is $\{8\}$