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Chapter 3: Problem 28

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$8\left(3^{6-x}\right)=40$$

Short Answer

Expert verified
Approximately 4.535 when rounded to three decimal places.

Step by step solution

01

Simplify the equation

Start by simplifying the equation. This can be achieved by dividing both sides of the equation by 8. Hence, the simplified equation becomes: \(3^{6-x} = \frac{40}{8} = 5\).
02

Apply the natural logarithm to both sides

By taking the natural logarithm (ln) of both sides of the equation, we can use the properties of logarithms to bring the variable x out of the exponent. This gives us: ln( \(3^{6-x}\)) = ln(5).
03

Use the properties of logarithms

Use the power rule of logarithms, which states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number. This changes the left side of the equation to: (6 - x) ln(3) = ln(5).
04

Solve for x

Isolate x on one side of the equation. First, divide both sides by ln(3) to give: 6 - x = \(\frac{ln(5)}{ln(3)}\). Then re-arrange to solve for x: x = 6 - \(\frac{ln(5)}{ln(3)}\). This is your final answer in algebraic form.
05

Approximate result

Substitute the values using a calculator in order to approximate the result to three decimal places: x = 6 - \(\frac{ln(5)}{ln(3)}\) \$\approx 6 - 1.465 = 4.535 .\)

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