Given that $\log _b 2 \approx 0.693, \log _b 3 \approx 1.099$, and $\log _b 5 \approx 1.609$, find each of the following, if possible. Round the answer to the nearest thousandth. $\\$$\log _b 15 b$

Find the logarithm using common logarithms and the change-of-base formula. $\\$$\log _3 20$

Convert to an exponential equation. $\\$$\log 7=0.845$

Students in an accounting class took a final exam and then took equivalent forms of the exam at monthly intervals thereafter. The average score $S(t)$, as a percent, after $t$ months was found to be given by the function

$$S(t)=78-15 \log (t+1), \quad t \geq 0 .$$

a) What was the average score when the students initially took the test, $t=0$ ?

b) What was the average score after $4$ months? after $24$ months?

The value of a stock is given by the function

$$V(t)=58\left(1-e^{-1.1 t}\right)+20,$$

where $V$ is the value of the stock after time $t$, in months. Find $V(1), V(2), V(4), V(6)$, and $V(12)$.

Use a graphing calculator to find the approximate solutions of the equation. $\\$$5 e^{5 x}+10=3 x+40$

Given that $\log _a 2=0.301, \log _a 5=0.699$, and $\log _a 6=0.778$, find each of the following. $\\$$\log _a 50$

Fiancé(e) Visas. The number of foreign nationals who came to the United States to marry an American using a "fiancé(e) visa" has grown exponentially in recent years. The total number of fiancé(e) visas issued is given by the function

$$f(x)=5728.98(1.1214)^x,$$

where $x$ is the number of years since 1990 (Source: U.S. Department of Homeland Security). Find the total number of fiancé(e) visas issued in 2006 and in 2009.

For each function: $\\$ a) Determine whether it is one-to-one.$

$$b) If the function is one-to-one, find a formula for the inverse.$$

f(x)=4 x^2+3, x \geq 0$

Find each of the following using a calculator. Round to four decimal places. $\\$$\ln (-4)$

The central processing unit (CPU) power in computers has increased significantly over the years. The CPU power in Macintosh computers has grown exponentially from $8 \mathrm{MHz}$ in 1984 to $3400 \mathrm{MHz}$ in 2013 (Source: Apple). The exponential function

$$M(t)=7.91477(1.26698)^t,$$

where $t$ is the number of years after 1984, can be used to estimate the CPU power in a Macintosh computer in a given year. Find the CPU power of a Macintosh Performa $5320$ $\mathrm{CD}$ in 1995 and of an iMac G6 in 2009. Round to the nearest one $\mathrm{MHz}$.

The number of U.S. troops diagnosed as overweight has more than doubled since 1998. The exponential function

$$W(x)=23,672.16(1.112)^x,$$

where $x$ is the number of years since 1998, which can be used to estimate the number of overweight active troops. Find the number of overweight troops in 2000, 2008, and in 2013.

Given that $\log _b 2 \approx 0.693, \log _b 3 \approx 1.099$, and $\log _b 5 \approx 1.609$, find each of the following, if possible. Round the answer to the nearest thousandth.

$\log _b \frac{3}{b}$

Given that $\log _a 2=0.301, \log _a 5=0.699$, and $\log _a 6=0.778$, find each of the following.

$$\log _a 3$$