We are given the equation:
[tex]\[5 \cdot \frac{5^{5x}}{25^x} = 25\][/tex]
We need to solve for [tex]\(x\)[/tex]. Let's proceed step-by-step:
1. Simplify the Fraction:
Recall that [tex]\(25 = 5^2\)[/tex]. So, we can rewrite [tex]\(25^x\)[/tex] as [tex]\((5^2)^x = 5^{2x}\)[/tex]. Substituting this in, the equation becomes:
[tex]\[
5 \cdot \frac{5^{5x}}{5^{2x}} = 25
\][/tex]
2. Simplify the Exponents:
Using the laws of exponents, [tex]\(\frac{5^{5x}}{5^{2x}} = 5^{5x - 2x} = 5^{3x}\)[/tex]. Thus, the equation reduces to:
[tex]\[
5 \cdot 5^{3x} = 25
\][/tex]
3. Combine Like Terms:
Since [tex]\(5 \cdot 5^{3x} = 5^{1 + 3x}\)[/tex], the equation now reads:
[tex]\[
5^{1 + 3x} = 25
\][/tex]
4. Express 25 as a power of 5:
Recall that [tex]\(25 = 5^2\)[/tex]. Substituting this in, we get:
[tex]\[
5^{1 + 3x} = 5^2
\][/tex]
5. Equate the Exponents:
Since the bases are the same, we can equate the exponents:
[tex]\[
1 + 3x = 2
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
Subtract 1 from both sides of the equation:
[tex]\[
3x = 1
\][/tex]
Divide both sides by 3:
[tex]\[
x = \frac{1}{3}
\][/tex]
So, one of the solutions is:
[tex]\[
x = \frac{1}{3}
\][/tex]
7. Considering Complex Solutions:
Since this is an exponential equation, there might be complex solutions as well. Exponential equations can have additional solutions involving complex numbers due to the periodic nature of the exponential function in the complex plane. The complete set of solutions includes:
[tex]\[
x = \frac{1}{3}, \quad x = \frac{\log(5) - 2i\pi}{3\log(5)}, \quad x = \frac{\log(5) + 2i\pi}{3\log(5)}
\][/tex]
Therefore, the solutions to the equation [tex]\(5 \cdot \frac{5^{5x}}{25^x} = 25\)[/tex] are:
[tex]\[
x = \frac{1}{3}, \quad x = \frac{\log(5) - 2i\pi}{3\log(5)}, \quad x = \frac{\log(5) + 2i\pi}{3\log(5)}
\][/tex]
