To solve the equation
[tex]\[
2\left(\frac{1}{49}\right)^{x-2} = 14,
\][/tex]
we need to isolate the variable [tex]\( x \)[/tex]. Here is a step-by-step approach:
1. Isolate the exponential expression:
Begin by dividing both sides of the equation by 2 to simplify:
[tex]\[
\left(\frac{1}{49}\right)^{x-2} = \frac{14}{2} = 7.
\][/tex]
2. Transform the exponential base:
Recall that [tex]\( \frac{1}{49} \)[/tex] can be written as [tex]\( 49^{-1} \)[/tex] and [tex]\( 49 \)[/tex] is equivalently [tex]\( 7^2 \)[/tex]:
[tex]\[
\left(49^{-1}\right)^{x-2} = 7.
\][/tex]
Simplify the left side:
[tex]\[
(7^{-2})^{x-2} = 7.
\][/tex]
Which simplifies further:
[tex]\[
7^{-2(x-2)} = 7.
\][/tex]
3. Equate the exponents (since the bases are the same):
The equation [tex]\( 7^{-2(x-2)} = 7^1 \)[/tex] implies that the exponents must be equal:
[tex]\[
-2(x-2) = 1.
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
First, distribute the [tex]\(-2\)[/tex]:
[tex]\[
-2x + 4 = 1.
\][/tex]
Next, rearrange to solve for [tex]\( x \)[/tex]:
[tex]\[
-2x = 1 - 4,
\][/tex]
[tex]\[
-2x = -3,
\][/tex]
[tex]\[
x = \frac{-3}{-2} = \frac{3}{2}.
\][/tex]
Therefore, the correct solution to the equation is [tex]\( \boxed{\frac{3}{2}} \)[/tex].
