Answer
$$x \approx 0.430618$$
Work Step by Step
$Use$ $a$ $calculator$ $to$ $find$ $the$ $solution$ $of$ $the$ $equation,$ $rounded$ $to$ $six$ $decimal$ $places:$
$$5^{-\frac{2x}{3}} = 0.63$$
Write the exponent form to logarithmic form: $b^c = a \rightarrow \log_b a =c $
$5^{-\frac{2x}{3}} = 0.63 \rightarrow \log_5 0.63 = -\frac{2x}{3}$
$\log_5 0.63 = -\frac{2x}{3}$
Solve for x
Multiply both sides by 3
$3(\log_5 0.63) = 3(\frac{2x}{3})$
$3\log_5 0.63 = 2x$
Divide both sides by -2
$\frac{3\log_5 0.63}{-2} = \frac{2x}{-2}$
$$x = \frac{3\log_5 0.63}{-2}$$
Plug into calculator
$$x = 0.430618158078972$$
Round to six decimal places
$$x \approx 0.430618$$