Certainly! Let's solve the equation [tex]\( 14^{10x} = 11^{x+10} \)[/tex] for [tex]\( x \)[/tex].
1. Take the natural logarithm (ln) of both sides of the equation to facilitate solving for the exponent.
[tex]\[
\ln(14^{10x}) = \ln(11^{x+10})
\][/tex]
2. Apply the logarithmic power rule [tex]\( \ln(a^b) = b \ln(a) \)[/tex] to bring the exponents down:
[tex]\[
10x \ln(14) = (x + 10) \ln(11)
\][/tex]
3. Expand the right-hand side:
[tex]\[
10x \ln(14) = x \ln(11) + 10 \ln(11)
\][/tex]
4. Rearrange the equation to isolate terms involving [tex]\( x \)[/tex] on one side:
[tex]\[
10x \ln(14) - x \ln(11) = 10 \ln(11)
\][/tex]
5. Factor [tex]\( x \)[/tex] out from the left-hand side:
[tex]\[
x (10 \ln(14) - \ln(11)) = 10 \ln(11)
\][/tex]
6. Divide both sides by [tex]\( (10 \ln(14) - \ln(11)) \)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{10 \ln(11)}{10 \ln(14) - \ln(11)}
\][/tex]
Therefore, the exact solution for [tex]\( x \)[/tex] using natural logarithms is:
[tex]\[
x = \frac{10 \ln(11)}{10 \ln(14) - \ln(11)}
\][/tex]
This concludes our detailed, step-by-step solution.
