Question

-Exponential and Logarithmic Functions-Solve the following. Round the answer to the nearest ten thousandth.

Answer 1

We want to find the solutions for the following equation

$7^{x+3}=27^{x+2}$

Using the following property

$\ln a^b=b\ln a$

we can apply the natural log on both sides of our equation and rewrite it

$\begin{gathered} 7^{x+3}=27^{x+2} \\ \ln 7^{x+3}=\ln 27^{x+2} \\ (x+3)\ln 7=(x+2)\ln 27 \end{gathered}$

Using the distributive property, we have

$\begin{gathered} (x+3)\ln 7=(x+2)\ln 27 \\ x\ln 7+3\ln 7=x\ln 27+2\ln 27 \end{gathered}$

Rewritting the expression isolating the unknown value, we have

$\begin{gathered} x\ln 7+3\ln 7=x\ln 27+2\ln 27 \\ x\ln 7-x\ln 27=2\ln 27-3\ln 7 \\ x(\ln 7-\ln 27)=2\ln 27-3\ln 7 \\ x=\frac{2\ln27-3\ln7}{\ln7-\ln27} \end{gathered}$

Now, using a calculator we can find our result.

$x=\frac{2\ln27-3\ln7}{\ln7-\ln27}=-0.55850682513\ldots\approx-0.5585$