To solve the equation \( 11^{-x + 7} = 5^{-3x} \), we’ll need to apply some logarithmic properties to isolate the variable \( x \). Here's the detailed, step-by-step solution:
1. Take the natural logarithm of both sides:
[tex]\[
\ln(11^{-x + 7}) = \ln(5^{-3x})
\][/tex]
2. Use the power rule of logarithms (\(\ln(a^b) = b \ln(a)\)):
[tex]\[
(-x + 7) \ln(11) = (-3x) \ln(5)
\][/tex]
3. Expand the equation:
[tex]\[
-x \ln(11) + 7 \ln(11) = -3x \ln(5)
\][/tex]
4. Isolate the terms involving \( x \):
[tex]\[
7 \ln(11) = -3x \ln(5) + x \ln(11)
\][/tex]
5. Factor \( x \) out on the right-hand side:
[tex]\[
7 \ln(11) = x (\ln(11) + 3 \ln(5))
\][/tex]
6. Solve for \( x \) by dividing both sides by \((\ln(11) + 3 \ln(5))\):
[tex]\[
x = \frac{7 \ln(11)}{\ln(11) + 3 \ln(5)}
\][/tex]
7. Evaluate the expression (note: this typically requires a calculator to get the numerical values of the logarithms):
[tex]\[
\ln(11) \approx 2.3979, \quad \ln(5) \approx 1.6094
\][/tex]
[tex]\[
x = \frac{7 \times 2.3979}{2.3979 + 3 \times 1.6094} = \frac{16.7853}{7.2261} \approx 2.322
\][/tex]
When solving exactly, recognizing constants derived from logarithms:
Thus the answer is approximately \( 2.0use \implies-8.5\approxNatural logarithm ratio \boxed{-7}
