91Ó°ÊÓ

Solve each equation and check your answers. $$\log (2 x-5)-\log 78=-1$$

pH level: \(f(x)=-\log _{10} x\) The pH level of a solution indicates the concentration of hydrogen \(\left(\mathrm{H}^{+}\right)\) ions in a unit called moles per liter. The pH level \(f(x)\) is given by the formula shown, where \(x\) is the ion concentration (given in scientific notation). A solution with \(\mathrm{pH}<7\) is called an acid (lemon juice: \(\mathrm{pH} \approx 2\) ), and a solution with \(\mathrm{pH}>7\) is called a base (household ammonia: \(\mathrm{pH} \approx 11\) ). Use the formula to determine the pH level of tomato juice if \(x=7.94 \times 10^{-5}\) moles per liter. Is this an acid or base solution?

Solve each equation using the uniqueness property of logarithms. $$\log _{3}(x+6)-\log _{3} x=\log _{3} 5$$

Graph each function \(f(x)\) and its inverse \(f^{-1}(x)\) on the same grid and "dash-in" the line \(y=x\). Note how the graphs are related. Then verify the "inverse function" relationship using a composition. $$f(x)=\sqrt[3]{x-7} ; f^{-1}(x)=x^{3}+7$$

Graph each function \(f(x)\) and its inverse \(f^{-1}(x)\) on the same grid and "dash-in" the line \(y=x\). Note how the graphs are related. Then verify the "inverse function" relationship using a composition. $$f(x)=\frac{2}{9} x+4 ; f^{-1}(x)=\frac{9}{2} x-18$$

Solve each equation using the uniqueness property of logarithms. $$\ln (x-1)+\ln 6=\ln (3 x)$$

Assuming the rate of inflation is \(5 \%\) per year, the predicted price of an item can be modeled by the function \(P(t)=P_{0}(1.05)^{t},\) where \(P_{0}\) represents the initial price of the item and \(t\) is in years. Use this information to solve . What will the price of a new car be in the year \(2010,\) if it cost \(\$ 20,000\) in the year \(2000 ?\)

Intensity of sound: The intensity of sound as perceived by the human ear is measured in units called decibels (dB). The loudest sounds that can be withstood without damage to the eardrum are in the 120 - to 130 -dB range, while a whisper may measure in the 15 - to 20 -dB range. Decibel measure is given by the equation \(D(I)=10 \log \left(\frac{I}{I_{0}}\right),\) where \(I\) is the actual intensity of the sound and \(I_{0}\) is the faintest sound perceptible by the human earcalled the reference intensity. The intensity \(I\) is often given as a multiple of this reference intensity, but often the constant \(10^{-16}\) (watts per cm \(^{2}\); \(\left.\mathrm{W} / \mathrm{cm}^{2}\right)\) is used as the threshold of audibility. Sound intensity of a hair dryer: Every morning (it seems), Jose is awakened by the mind-jarring, ear-jamming sound of his daughter's hair dryer \((75 \mathrm{dB}) .\) He knew he was exaggerating, but told her (many times) of how it reminded him of his railroad days, when the air compressor for the pneumatic tools was running \((110 \mathrm{dB}) .\) In fact, how many times more intense was the sound of the air compressor compared to the sound of the hair dryer?

Write as many of the following formulas as you can from memory: a. perimeter of a rectangle b. area of a circle c. volume of a cylinder d. volume of a cone e. circumference of a circle f. area of a triangle g. area of a trapezoid h. volume of a sphere i. Pythagorean theorem

Stocking a lake: A farmer wants to stock a private lake on his property with catfish. A specialist studies the area and depth of the lake, along with other factors, and determines it can support a maximum population of around 750 fish, with growth modeled by the function \(P(t)=\frac{750}{1+24 e^{-0.075 t}},\) where \(P(t)\) gives the current population after \(t\) months. (a) How many catfish did the farmer initially put in the lake? (b) How many months until the population reaches 300 fish?