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Chapter 12: Problem 14

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\ln \left(\frac{e^{4}}{8}\right)$$

Short Answer

Expert verified
The expanded form of the expression \(\ln \left(\frac{e^{4}}{8}\right)\) is 4 鈭� \ln(8).

Step by step solution

01

Identify the parts of the logarithmic expression

Firstly, identify that you're given a natural log function (\ln) of a fraction. This fraction can be split up into two separate logarithms through the rule: \ln(a / b) = \ln(a) - \ln(b). The fraction is \(\frac{e^4}{8}\), so \(a = e^4\) and \(b = 8\).\n
02

Apply the Logarithm Division Rule

Next, use the logarithm division rule mentioned above to split the function into two parts: \ln(e^4)-\ln(8). Now your expression is simplified and can be executed separately.\n
03

Simplify \ln(e^4)

The power rule for logarithms can be expressed as \ln(a^n)= n* \ln(a). Thus, \ln(e^4)= 4* \ln(e). As the logarithm base \(e\) of \(e\) is 1, therefore \ln(e^4)=4.\n
04

Evaluate \ln(8)

The value \ln(8) cannot be simplified further without a calculator because 8 is not a power of \(e\). Thus, we leave it as is.\n
05

Combine the results

Now we combine the results from Step 3 and Step 4. The final expanded form of the original expression is now 4 - \ln(8).\n

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Most popular questions from this chapter

I can solve \(4^{x}=15\) by writing the equation in logarithmic form.

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 _______.

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. $$5^{x}=3 x+4$$

$$\text { Solve: } \frac{x+2}{4 x+3}=\frac{1}{x}$$

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. Use the formula to solve Exercises. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?
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