Chapter 4

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \text { If } x=\frac{1}{k} \ln y, \text { then } y=e^{k x} $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Examples of exponential equations include \(10^{x}=5.71\) \(e^{x}=0.72,\) and \(x^{10}=5.71\)

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)

If \(\$ 4000\) is deposited into an account paying \(3 \%\) interest compounded annually and at the same time \(\$ 2000\) is deposited into an account paying \(5 \%\) interest compounded annually, after how long will the two accounts have the same balance? Round to the nearest year.

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40\)

Check each proposed solution by direct substitution or with a graphing utility. $$ (\ln x)^{2}=\ln x^{2} $$

This group exercise involves exploring the way we grow. - Group members should create a graph for the function that models the percentage of adult height attained by a boy who is \(x\) years old, \(f(x)=29+48.8 \log (x+1) .\) Let \(x=5,6\) \(7, \ldots ., 15,\) find function values, and connect the resulting points with a smooth curve. Then create a graph for the function that models the percentage of adult height attained by a girl who is \(x\) years old, \(g(x)=62+35 \log (x-4)\) Let \(x=5,6,7, \ldots, 15,\) find function values, and connect the resulting points with a smooth curve. Group members should then discuss similarities and differences in the growth patterns for boys and girls based on the graphs.

Check each proposed solution by direct substitution or with a graphing utility. $$ (\log x)(2 \log x+1)=6 $$

Three of the richest comedians in the United States are Larry David (creator of Seinfeld), Matt Groening (creator of The Simpsons), and Trey Parker (co- creator of South Park). Larry David is worth \(\$ 450\) million more than Trey Parker. Matt Groening is worth \(\$ 150\) million more than Trey Parker. Combined, the net worth of these three comedians is \(\$ 1650\) million (or \(\$ 16.5\) billion). Determine how much, in millions of dollars, each of these comedians is worth. (Source: petamovies.com) (Section 1.3, Example 1)

Check each proposed solution by direct substitution or with a graphing utility. $$ \ln (\ln x)=0 $$

If \(f(x)=m x+b,\) find \(\frac{f(x+h)-f(x)}{h}, h \neq 0\) (Section \(2.2,\) Example 8 )

Research applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: pH (acidity of solutions), intensity of sound (decibels), brightness of stars, human memory, progress over time in a sport, profit over time. For the area that you select, explain how logarithmic functions are used and provide examples.

Find the inverse of \(f(x)=x^{2}+4, x \geq 0\) (Section \(2.7, \text { Example } 7)\)

Exercises 150–152 will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 32\) b. Evaluate: \(\log _{2} 8+\log _{2} 4\) c. What can you conclude about \(\log _{2} 32,\) or \(\log _{2}(8 \cdot 4) ?\)

Consider the quadratic function $$f(x)=-4 x^{2}-16 x+3$$ a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. (Section 3.1, { Example } 4)

Exercises 150–152 will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about $$ \log _{2} 16, \text { or } \log _{2}\left(\frac{32}{2}\right) ? $$

Solve the equation \(x^{3}-9 x^{2}+26 x-24=0\) given that 4 is a zero of \(f(x)=x^{3}-9 x^{2}+26 x-24\) Example \(6)\)

Exercises 150–152 will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$ \log _{3} 81, \text { or } \log _{3} 9^{2} ? $$

Exercises \(153-155\) will help you prepare for the material covered in the next section. a. Simplify: \(e^{\ln 3}\). b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)