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 \\ (x(-3-2ln33))/((-3-2ln33))=(8ln33+3)/((-3-2ln33)) \\ x=(8ln33+3)/((-3-2ln33)) \end{gathered}`

Answer;

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