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What is the solution to this equation?

[tex] 2\left(\frac{1}{49}\right)^{x-2}=14 [/tex]

A. 3
B. [tex] -\frac{5}{2} [/tex]
C. [tex] \frac{3}{2} [/tex]
D. [tex] \frac{5}{2} [/tex]


Sagot :

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].