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Chapter 4: Problem 65

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln \sqrt{x+3}=1$$

Short Answer

Expert verified
The exact answer to the equation is \(x = e^2 - 3\) and the decimal approximation for this solution, rounded to two decimal places, will need a calculator depending upon the precision of \(e\). The domain of the answer is valid.

Step by step solution

01

Rewrite the logarithmic equation

The logarithmic equation can be rewritten as an equivalent exponential equation. In this case, \(\ln \sqrt{x+3}=1\) is equivalent to \(e^1 = \sqrt{x+3}\) due to the nature of natural logs being the logarithm to the base \(e\). Hence the equation results in \(e = \sqrt{x+3}\).
02

Square both sides

To eliminate the square root, square both sides of the equation. This gives \((e)^2 = (\sqrt{x+3})^2\), which simplifies to \(e^2 = x+3\).
03

Solve for \(x\)

To solve for \(x\), subtract 3 from both sides of the equation. This gives \(x = e^2 -3\). Now, compute the actual decimal value for the right hand side expression to get the numerical value for \(x\).
04

Verify domain

Lastly, we must validate that the solution of \(x\) lies within the domain of the original logarithmic expression. It does, because \(x\) is less than -3, which is the boundary for the original log's domain. Therefore, the solution is valid.

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Most popular questions from this chapter

In parts (a)-(c), graph \(f\) and \(g\) in the same viewing rectangle. a. \(f(x)=\ln (3 x), g(x)=\ln 3+\ln x\) b. \(f(x)=\log \left(5 x^{2}\right), g(x)=\log 5+\log x^{2}\) c. \(f(x)=\ln \left(2 x^{3}\right), g(x)=\ln 2+\ln x^{3}\) d. Describe what you observe in parts (a)-(c). Generalize this observation by writing an equivalent expression for \(\log _{b}(M N),\) where \(M>0\) and \(N>0\) e. Complete this statement: The logarithm of a product is equal to _________.

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. \(y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}\)

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$2^{x+1}=8$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations $$\log (3 x+1)=5 \text { and } \log (3 x+1)=\log 5$$ are similar, I solved them using the same method.
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