91Ó°ÊÓ

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \ln (5 x)+\ln 1=\ln (5 x) $$

The loudness level of a sound, D, in decibels, is given by the formula $$ D=10 \log \left(10^{12} I\right) $$ where I is the intensity of the sound, in watts per meter2. Decibel levels range from 0, a barely audible sound, to 160, a sound resulting in a ruptured eardrum. (Any exposure to sounds of 130 decibels or higher puts a person at immediate risk for hearing damage.) Use the formula to solve Exercises 117–118. The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter? Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?

The function \(P(x)=95-30 \log _{2} x\) models the percentage, \(P(x),\) of students who could recall the important features of a classroom lecture as a function of time, where \(x\) represents the number of days that have elapsed since the lecture was given. The figure at the top of the next column shows the graph of the function. Use this information to solve Exercises \(117-118\). Round answers to one decimal place. After how many days have all students forgotten the important features of the classroom lecture? (Let \(P(x)=0\) and solve for \(x\).) Locate the point on the graph that conveys this information.

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can use any positive number other than 1 in the change-of-base property, but the only practical bases are 10 and \(e\) because my calculator gives logarithms for these two bases.

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

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use this function to solve. Use an equation to answer this question: How far from the eye of a hurricane is the barometric air pressure 29 inches of mercury? Use the TRACE and ZOOM features or the intersect command of your graphing utility to verify your answer.

Use the Leading Coefficient Test to determine the end behavior of the graph of \(f(x)=-2 x^{2}(x-3)^{2}(x+5)\) (Section \(3.2,\) Example 2 )

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.