Question

solve the following equation. express your answer as both an exact expression and a decimal approximation rounded to two decimal places. use e = 2.71828182845905.

Answer 1

SOLUTION:

Case: Exponential equations

Given:

$e^{3x-8}=102^{5x+5}$

Required: To solve for x

$\begin{gathered} e^{3x-8}=102^{5x+5} \\ Take\text{ }natural\text{ }logarithm\text{ }of\text{ }both\text{ }sides \\ ln(e^{3x-8})=ln(102^{5x+5}) \\ (3x-8)lne^=(5x+5)ln(102) \\ 3x-8=(5x+5)4.624972 \\ 3x-8=23.12x+23.12 \\ Collecting\text{ }like\text{ }terms \\ -23.12-8=23.12x-3x \\ -31.12=20.12x \\ x=-\frac{31.12}{20.12} \\ x=-1.54658754284 \end{gathered}$

Final answer:

Exact value

$x=-1.54658754284$

To 2 decimal places

$x\approx-1.55$