Answer:
Step-by-step explanation:
To solve this question, we can use either of the 2 below given methods:
- Rule of exponents
- Logarithms
1. Rule of Exponents:
[tex]2 ^ { 7-2x } = \frac{ 1 }{ 4 }\\\rightarrow 2 ^ { 7-2x } = \frac{ 1 }{ 2^{2} }[/tex]
Now, by using the law → [tex]x^{-y} = \frac{1}{x^{y}}[/tex]...
[tex]2 ^ { 7-2x } = \frac{ 1 }{ 2^{2} }\\2 ^ { 7-2x } = 2^{-2}[/tex]
Now, let's take the exponential values as the base values are equal.
[tex]7 - 2x = - 2\\- 2x = - 2 + (-7)\\- 2x = - 9\\\boxed{x = \frac{9}{2} = 4.5}[/tex]
2. Logarithms:
[tex]2 ^ { 7-2x } = \frac{ 1 }{ 4 }\\\rightarrow2^{-2x+7}=\frac{1}{4}[/tex]
Now, take the logarithm of both the sides of the equation.
[tex]\log(2^{-2x+7})=\log(\frac{1}{4})[/tex]
We know that, the logarithm of a number raised to an exponential power is power times the logarithm of the number. So,
[tex]\log(2^{-2x+7})=\log(\frac{1}{4}) \\ \rightarrow \left(-2x+7\right)\log(2)=\log(\frac{1}{4})[/tex]
Now, divide both the sides of the equstion by log (2).
[tex]-2x+7=\frac{\log(\frac{1}{4})}{\log(2)}[/tex]
According to the change of base formula, [tex]\frac{\log(x)}{\log(y)}[/tex] = [tex]\log_{y}(x)[/tex]. Then,
[tex]-2x+7=\log_{2}\left(\frac{1}{4}\right)[/tex]
By subtracting 7 from both the sides of the equation & then simplifing it further....
[tex]-2x=-2-7 \\-2x = - 9\\\boxed{x = \frac{9}{2} = 4.5}[/tex]
- We get the same value by using either of the 2 methods.
- The value of x = 9/2 or 4.5
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Hope it helps!
[tex]\mathfrak{Lucazz}[/tex]
