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Chapter 3: Problem 67

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$3-\ln x=0$$

Short Answer

Expert verified
The rounded solution to three decimal places is \(x = 20.086\).

Step by step solution

01

Isolate the Logarithm

Rearrange the equation to isolate the logarithm. We subtract 3 from each side to isolate the logarithmic term, which gives us: \(-\ln x = -3\). Now, multiply each side by -1 to get the logarithm alone on one side: \(\ln x = 3\).
02

Convert Logarithm to Exponential Form

Next, we convert the logarithmic equation to exponential form. From the definition of natural logarithms, we know that if \(\ln a = b\), then \(e^b = a\). Thus, \(\ln x = 3 \) becomes \(e^3 = x \).
03

Solve the Equation Numerically and Graphically

To solve the equation numerically, we calculate \(e^3 \) to get an approximate value of x. For the graphical solution, we graph the equation and find the point where the graph crosses the x-axis. This will also give us the approximate value of x.
04

Verify the Result

To verify our solution, we substitute \(x = e^3\) into the original equation \(3-\ln x=0\). The left side of the equation should equal the right side if our solution \(x = e^3\) is correct.
05

Round to Three Decimal Places

Since the problem wants an answer to three decimal places, we will round our values to three decimal places.

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln (x+1)-\ln (x-2)=\ln x$$

Function \(\quad\) Value $$ g(x)=-\ln x \quad x=\frac{1}{2}$$

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 the functions \(y_{1}=\ln x-\ln (x-3)\) and \(y_{2}=\ln \frac{x}{x-3}\) in the same viewing window. Does the graphing utility show the functions with the same domain? If so, should it? Explain your reasoning.

You invest \(\$ 2500\) in an account at interest rate \(r,\) compounded continuously. Find the time required for the amount to (a) double and (b) triple. $$r=0.0375$$
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