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Chapter 3: Problem 34

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$8\left(4^{6-2 x}\right)+13=41$$

Short Answer

Expert verified
After calculating, x approximates to 0.792.

Step by step solution

01

Reduce the Equation

By subtracting 13 from both sides, it gets: 8\(4^{6-2 x}\) = 41 - 13, hence 8(4^{6-2x}) = 28.
02

Simplify the Equation

Divide the equation by 8 to isolate the exponential function: \(4^{6-2 x}\) = 28/8, hence \(4^{6-2x}\) = 3.5.
03

Apply Natural Logarithm

We take the natural logarithm (in base e) of both sides. Due to the nature of the logarithm and exponentiation, the expression simplifies further: ln(\(4^{6-2 x}\)) = ln(3.5), hence (6 - 2x)ln(4) = ln(3.5).
04

Solve for x

Now that we have a simpler linear equation, we can solve for x: x = (ln(3.5)/ln(4) - 6) / -2.

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Most popular questions from this chapter

The values \(y\) (in billions of dollars) of U.S. currency in circulation in the years 2000 through 2010 can be modeled by \(y=-611+507\) ln \(t, 10 \leq t \leq 20\) where \(t\) represents the year, with \(t=10\) corresponding to 2000. During which year did the value of U.S. currency in circulation exceed \(\$ 690\) billion? (Source: Board of Governors of the Federal Reserve System )

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