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Chapter 3: Problem 30

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$1000 e^{-4 x}=75$$

Short Answer

Expert verified
The solution to the exponential equation to three decimal places is \(x \approx 1.386\).

Step by step solution

01

Isolate the exponential term

Divide both sides of the equation by 1000 to isolate the exponential term on one side: \[-4x = \frac{75}{1000}\] which simplifies to \[-4x = 0.075\]
02

Apply natural logarithm

Apply the natural logarithm (ln) to both sides to get rid of the base e on the left side: \[ln(-4x) = ln(0.075)\]
03

Use logarithm rules

Use the logarithm rule that the logarithm of a product is equal to the sum of the logarithms \[ln(-4) + ln(x) = ln(0.075)\]
04

Solve for x

Isolate x on one side using basic algebra: \[x = \frac{ln(0.075) - ln(-4)}{-4}\]
05

Evaluate the expression

Use a calculator to find the numerical values of ln(0.075) and ln(-4), divide the result by -4 and round to three decimal places to get the final answer.

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Most popular questions from this chapter

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. $$x=e^{-5 / 2}$$

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph.\( \)y=\log _{5}(x-1)+4$$

Function \(\quad\) Value $$f(x)=3 \ln x \quad x=0.74$$

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$f(x)=\ln (x-4)$$

Write the logarithmic equation in exponential form. $$\ln 250=5.521 \ldots$$
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