To find the value of [tex]\( x \)[/tex] in the equation [tex]\( 2 \left(5^x\right) = 14 \)[/tex], follow the steps below:
1. Isolate the Exponential Term:
First, divide both sides of the equation by 2 to isolate the exponential term:
[tex]\[
5^x = \frac{14}{2}
\][/tex]
Simplifying the right-hand side, we get:
[tex]\[
5^x = 7
\][/tex]
2. Apply the Logarithm:
To solve for [tex]\( x \)[/tex], take the logarithm of both sides. While any logarithm base can be used, the common logarithm (base 10) or the natural logarithm (base [tex]\( e \)[/tex]) are typically used. We'll use the common logarithm (base 10):
[tex]\[
\log(5^x) = \log(7)
\][/tex]
3. Use the Power Rule of Logarithms:
Use the logarithmic identity that states [tex]\( \log(a^b) = b \log(a) \)[/tex]:
[tex]\[
x \log(5) = \log(7)
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by dividing both sides of the equation by [tex]\( \log(5) \)[/tex]:
[tex]\[
x = \frac{\log(7)}{\log(5)}
\][/tex]
Therefore, the correct value for [tex]\( x \)[/tex] is:
[tex]\[
\boxed{\frac{\log 7}{\log 5}}
\][/tex]
Hence, the correct answer is:
A. [tex]\(\frac{\log 7}{\log 5}\)[/tex]
