To solve the equation [tex]\( 5^{x-2} + 5^x = 100 \)[/tex], we will break it down into clear, step-by-step processes. Here’s how we can approach it:
1. Rewrite the Equation:
First, notice that [tex]\( 5^{x-2} \)[/tex] can be written in terms of [tex]\( 5^x \)[/tex]. This helps us consolidate the base terms.
[tex]\[
5^{x-2} = \frac{5^x}{5^2} = \frac{5^x}{25}
\][/tex]
So the equation transforms to:
[tex]\[
\frac{5^x}{25} + 5^x = 100
\][/tex]
2. Combine Like Terms:
Factor out [tex]\( 5^x \)[/tex] from the left-hand side:
[tex]\[
\frac{5^x}{25} + 5^x = 100
\][/tex]
Rewrite the equation:
[tex]\[
\frac{5^x}{25} + 5^x = 100
\][/tex]
Combine the like terms:
[tex]\[
5^x \left( \frac{1}{25} + 1 \right) = 100
\][/tex]
3. Simplify Inside the Parentheses:
Simplify the expression inside the parentheses:
[tex]\[
\frac{1}{25} + 1 = \frac{1}{25} + \frac{25}{25} = \frac{26}{25}
\][/tex]
So, the equation now becomes:
[tex]\[
5^x \cdot \frac{26}{25} = 100
\][/tex]
4. Isolate [tex]\( 5^x \)[/tex]:
To isolate [tex]\( 5^x \)[/tex], multiply both sides by [tex]\( \frac{25}{26} \)[/tex]:
[tex]\[
5^x = 100 \cdot \frac{25}{26} = \frac{2500}{26} = \frac{1250}{13}
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Take the logarithm of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x \log(5) = \log\left(\frac{1250}{13}\right)
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{\log\left(\frac{1250}{13}\right)}{\log(5)}
\][/tex]
The final step gives us the required solution. Therefore, the solution for [tex]\( x \)[/tex] is:
[tex]\[
x = \frac{\log\left(1250\right) - \log(13)}{\log(5)}
\][/tex]
