Solution
Step 1:
Write the equation
[tex]\frac{2}{5}\text{ }\cdot\text{ 4}^{5x}\text{ - 8 = 4}[/tex]Step 2:
Multiply each term by 5 to cancel out the denominator.
[tex]\begin{gathered} \frac{2}{5}\operatorname{\cdot}(\text{4})^{5x}\text{- 8 = 4} \\ 5\times\frac{2}{5}\operatorname{\cdot}(\text{4})^{5x}\text{-5}\times\text{8 = 5}\times\text{4} \\ 2\operatorname{\cdot}(\text{4})^{5x}\text{- 40 = 20} \end{gathered}[/tex]Step 3:
Divide through by 2
[tex]\begin{gathered} 2\operatorname{\cdot}\text{ \lparen4\rparen}^{5x}\text{ - 40 = 20} \\ \frac{2\operatorname{\cdot}\text{ \lparen4\rparen}^{5x}}{2}\text{ - }\frac{40}{2}\text{ = }\frac{20}{2} \\ 4^{5x}\text{ - 20 = 10} \\ 4^{5x}\text{ = 10 + 20} \\ 4^{5x}\text{ = 30} \end{gathered}[/tex]Step 4:
Take the natural logarithm of both sides
[tex]\begin{gathered} 4^{5x}\text{ = 30} \\ ln(4)^{5x}\text{ = ln\lparen30\rparen} \\ \text{5x ln\lparen4\rparen = ln\lparen30\rparen} \\ \text{5x = }\frac{ln(30)}{ln(4)} \\ \text{5x = 2.453445298} \\ \text{ x = }\frac{2.453445298}{5} \\ x=0.49068 \end{gathered}[/tex]Final answer
x = 0.49068