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Chapter 3: Problem 50

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

Short Answer

Expert verified
The solution to the equation is \(x = 6 - log_35\), and when approximated to three decimal places, it equals \(x \approx 3.465\).

Step by step solution

01

Isolate the exponential term

Divide both sides of the equation by 8 to isolate \((3^{6 - x})\) on one side of the equation. This gives \(3^{6 - x} = \frac{40}{8} = 5\).
02

Convert to common base

In order to apply the properties of exponents, convert 5 into the base of 3. \(5 = 3^{log_35}\). This gives \(3^{6 - x} = 3^{log_35}\) .
03

Apply the principle of exponent

Now that the bases are equal, the exponents should also be equal. Hence, the exponential equation becomes a simple algebraic equation \(6 - x = log_35\).
04

Solve for x

To solve for x, isolate x on one side of the equation. This will give \(x = 6 - log_35\).
05

Approximate the result

Use a calculator to find the approximate value of \(log_35\) and subtract it from 6. Round the answer to three decimal places as requested in the question.

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