Question

Solve the following equation for x using the natural logarithm

Answer 1

Use log rule
log(ab) = log a + log b

Take log of both sides:
$ln 4 + ln 4^x = ln 10 + ln e^{-x}$

Use log rule
log(a^n) = n*log(a)

$ln 4 + xln 4 = ln 10 -xln e \\ \\ ln 4 + xln 4 = ln 10 -x \\ \\ x ln 4+x = ln 10 - ln 4 \\ \\ x(ln 4 +1) = ln 10 -ln 4 \\ \\ x = \frac{ln 10 - ln 4}{ln 4 +1}$