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Marijn Pessers · Answer 1 · 2023-02-20T20:30:53+0000

Solution

Step 1:

Write the equation

$(2)/(5)\text{ }\cdot\text{ 4}^(5x)\text{ - 8 = 4}$

Step 2:

Multiply each term by 5 to cancel out the denominator.

$\begin{gathered} (2)/(5)\operatorname{\cdot}(\text{4})^(5x)\text{- 8 = 4} \\ 5*(2)/(5)\operatorname{\cdot}(\text{4})^(5x)\text{-5}*\text{8 = 5}*\text{4} \\ 2\operatorname{\cdot}(\text{4})^(5x)\text{- 40 = 20} \end{gathered}$

Step 3:

Divide through by 2

$\begin{gathered} 2\operatorname{\cdot}\text{ \lparen4\rparen}^(5x)\text{ - 40 = 20} \\ \frac{2\operatorname{\cdot}\text{ \lparen4\rparen}^(5x)}{2}\text{ - }(40)/(2)\text{ = }(20)/(2) \\ 4^(5x)\text{ - 20 = 10} \\ 4^(5x)\text{ = 10 + 20} \\ 4^(5x)\text{ = 30} \end{gathered}$

Step 4:

Take the natural logarithm of both sides

$\begin{gathered} 4^(5x)\text{ = 30} \\ ln(4)^(5x)\text{ = ln\lparen30\rparen} \\ \text{5x ln\lparen4\rparen = ln\lparen30\rparen} \\ \text{5x = }(ln(30))/(ln(4)) \\ \text{5x = 2.453445298} \\ \text{ x = }(2.453445298)/(5) \\ x=0.49068 \end{gathered}$

Final answer

x = 0.49068

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