Question

exponential equations
59pm points 100 submitting an external tool
current objective
use logarithms to solve exponential equations
question
find the exact solution to $15^{x - 11}=11^{x}$, or the solution rounded to the nearest tenth.
provide your answer below:
$x=$

Explanation:

Step1: Take the natural logarithm of both sides

$$\ln(15^{x - 11})=\ln(11^{x})$$
Using the property $\ln(a^{b})=b\ln(a)$, we get $(x - 11)\ln(15)=x\ln(11)$.

Step2: Expand the left - hand side

$$x\ln(15)-11\ln(15)=x\ln(11)$$

Step3: Move the terms with \(x\) to one side

$$x\ln(15)-x\ln(11)=11\ln(15)$$
Factor out \(x\): \(x(\ln(15)-\ln(11)) = 11\ln(15)\)

Step4: Solve for \(x\)

Using the property \(\ln(a)-\ln(b)=\ln(\frac{a}{b})\), we have \(x=\frac{11\ln(15)}{\ln(15)-\ln(11)}\)
Now, calculate the numerical value:
\(\ln(15)\approx2.70805\), \(\ln(11)\approx2.397895\)
\(x=\frac{11\times2.70805}{2.70805 - 2.397895}=\frac{29.78855}{0.310155}\approx96.0\)

Answer:

\(x = 96.0\)