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Chapter 3: Problem 32

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$-14+3 e^{x}=11$$

Short Answer

Expert verified
The approximate solution to the equation is \(x \approx 2.123\) to three decimal places.

Step by step solution

01

Isolate the Exponential Term

To isolate the exponential term, add 14 to both sides of the equation. This will give: \(3e^x = 25\)
02

Divide by 3

To simplify and further isolate the x, divide every term in the equation by 3 to obtain the equation \(e^x = \frac{25}{3}\).
03

Applying the Natural Logarithm

Apply the natural logarithm to both sides of the equation to counteract the exponential and solve for x. Using a property of logarithms, \(ln(e^x) = x\), this transforms the equation to \(x = \ln\left(\frac{25}{3}\right)\).
04

Approximate x to Three Decimal Places

We then use a calculator to compute \(x = \ln\left(\frac{25}{3}\right)\) and round the result to three decimal places. Hence \(x \approx 2.123\).

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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. $$x=e^{-5 / 6}$$

Writing a Natural Logarithmic Equation In Exercises \(53-56,\) write the exponential equation in logarithmic form. $$e^{1 / 2}=1.6487 \ldots$$

Using the One-to-One Property In Exercises \(73-76,\) use the One-to-One Property to solve the equation for \(x\). $$\ln (x+4)=\ln 12$

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window. \(f(x)=\ln (x-1)\)

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$8 e^{-2 x / 3}=11$$
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