Certainly! Let's walk through the process of solving the equation step-by-step:
We are given the equation:
[tex]\[
2\left(\frac{1}{49}\right)^{x-2} = 14
\][/tex]
1. Isolate the exponential term:
First, divide both sides of the equation by 2 to isolate the exponential expression:
[tex]\[
\left(\frac{1}{49}\right)^{x-2} = 7
\][/tex]
2. Rewrite the exponential term:
Recall that [tex]\(\frac{1}{49}\)[/tex] can be written as [tex]\(49^{-1}\)[/tex]. So the equation becomes:
[tex]\[
(49^{-1})^{x-2} = 7
\][/tex]
Which simplifies to:
[tex]\[
49^{-(x-2)} = 7
\][/tex]
3. Express 7 as a power of 49:
We recognize that [tex]\(7^2 = 49\)[/tex]. Therefore, [tex]\(7 = 49^{1/2}\)[/tex]. Substituting this into the equation:
[tex]\[
49^{-(x-2)} = 49^{1/2}
\][/tex]
4. Set the exponents equal to each other:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
-(x-2) = \frac{1}{2}
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Simplify the equation to get:
[tex]\[
-(x - 2) = \frac{1}{2}
\][/tex]
Distribute the negative sign:
[tex]\[
-x + 2 = \frac{1}{2}
\][/tex]
Subtract 2 from both sides:
[tex]\[
-x = \frac{1}{2} - 2
\][/tex]
Simplify the right-hand side:
[tex]\[
-x = \frac{1}{2} - \frac{4}{2} = -\frac{3}{2}
\][/tex]
Finally, multiply both sides by -1:
[tex]\[
x = \frac{3}{2}
\][/tex]
Thus, the solution to the equation is:
[tex]\[
x = \frac{3}{2}
\][/tex]
Therefore, the correct answer is [tex]\(\boxed{\frac{3}{2}}\)[/tex].
