Sure, I'd be happy to help you with a step-by-step solution to solve the equation [tex]\(7^{3-x} = 5^{x+1}\)[/tex].
1. Understand the Equation: The equation is [tex]\(7^{3-x} = 5^{x+1}\)[/tex]. We need to solve for [tex]\(x\)[/tex].
2. Express the Equation in Logarithmic Form:
- [tex]\(7^{3-x} = 5^{x+1}\)[/tex] can be rewritten using logarithms. Taking the natural logarithm (or log base 10) on both sides gives us:
[tex]\[
\log(7^{3-x}) = \log(5^{x+1})
\][/tex]
3. Apply Logarithm Properties:
- Use the logarithmic power rule [tex]\(\log(a^b) = b\log(a)\)[/tex]:
[tex]\[
(3-x) \log(7) = (x+1) \log(5)
\][/tex]
4. Distribute the Logarithms:
- Distribute [tex]\(\log(7)\)[/tex] and [tex]\(\log(5)\)[/tex] to the terms inside the parentheses:
[tex]\[
3 \log(7) - x \log(7) = x \log(5) + \log(5)
\][/tex]
5. Rearrange the Terms to Isolate [tex]\(x\)[/tex]:
- Move the terms involving [tex]\(x\)[/tex] to one side and constant terms to the other side:
[tex]\[
3 \log(7) - \log(5) = x \log(5) + x \log(7)
\][/tex]
- Factor out [tex]\(x\)[/tex] on the right side:
[tex]\[
3 \log(7) - \log(5) = x (\log(5) + \log(7))
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
- Divide both sides by [tex]\(\log(5) + \log(7)\)[/tex]:
[tex]\[
x = \frac{3 \log(7) - \log(5)}{\log(5) + \log(7)}
\][/tex]
7. Simplify Using Properties of Logarithms:
- Recognize that [tex]\(\log(5) + \log(7) = \log(35)\)[/tex]:
[tex]\[
x = \frac{3 \log(7) - \log(5)}{\log(35)}
\][/tex]
- Combine the terms in the numerator:
[tex]\[
x = \log_{35}\left(\frac{7^3}{5}\right)
\][/tex]
8. Final Expression:
- Simplify the fraction inside the logarithm:
[tex]\[
x = \log_{35}\left(\frac{343}{5}\right)
\][/tex]
Based on the final expression, we obtain:
[tex]\[
x = \log_{35}\left(\frac{343}{5}\right)
\][/tex]
Hence, the solution to the equation [tex]\(7^{3-x} = 5^{x+1}\)[/tex] is:
[tex]\[
x = \log_{35}\left(\frac{343}{5}\right)
\][/tex]
