To solve the equation [tex]\( 2^{5x - 1} = 3^x \)[/tex], follow these steps:
1. Understand the nature of the equation:
We have an equation involving exponential functions with different bases, [tex]\(2\)[/tex] and [tex]\(3\)[/tex].
2. Transform the equation using logarithms:
To handle the exponents with different bases, we apply logarithms (natural logarithms, for simplicity):
[tex]\[
\ln(2^{5x - 1}) = \ln(3^x)
\][/tex]
3. Use logarithmic properties to simplify:
Apply the property of logarithms that allows you to bring down the exponent:
[tex]\[
(5x - 1) \ln(2) = x \ln(3)
\][/tex]
4. Rearrange to isolate [tex]\(x\)[/tex]:
Let's distribute the logarithm terms and solve for [tex]\(x\)[/tex]:
[tex]\[
5x \ln(2) - \ln(2) = x \ln(3)
\][/tex]
[tex]\[
5x \ln(2) - x \ln(3) = \ln(2)
\][/tex]
5. Factor [tex]\(x\)[/tex] from the left-hand side:
[tex]\[
x (5 \ln(2) - \ln(3)) = \ln(2)
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
Divide both sides of the equation by [tex]\((5 \ln(2) - \ln(3))\)[/tex]:
[tex]\[
x = \frac{\ln(2)}{5 \ln(2) - \ln(3)}
\][/tex]
7. Express in a simpler form:
Recognize that the solution can be expressed in terms of another logarithm. Here, we simplify:
[tex]\[
x = \log_{32/3}(2)
\][/tex]
Note that [tex]\(\log_{32/3}(2)\)[/tex] can be interpreted using the change of base formula, but it effectively captures the solution in a compact form.
The precise solution is:
[tex]\[
x = \log_{32/3}(2)
\][/tex]
