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]
