Question

Solve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places. Use e= 2.71828182845905.

Answer 1

Step 1

Given;

$e^{-3x-3}=33^{2x+8}$

Required; Solve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places.

Step 2

Solve the equation

$\begin{gathered} Add\text{ natural logarithm to both sides} \\ lne^{-3x-3}=ln33^{2x+8} \\ -3x-3=(2x+8)(ln33) \end{gathered}$$\begin{gathered} -3x=2xln33+8ln33+3 \\ -3x-2xln33=8ln33+3 \\ x(-3-2ln33)=8ln33+3 \\ \frac{x(-3-2ln33)}{(-3-2ln33)}=\frac{8ln33+3}{(-3-2ln33)} \\ x=\frac{8ln33+3}{(-3-2ln33)} \end{gathered}$

Answer;

$\begin{gathered} As\text{ an exact expression, x =}\frac{8ln(33)+3}{-3-2ln(33)} \\ Approximation\text{ to 2 decimal places=-3.10} \end{gathered}$