Use a graphing calculator to find the approximate solutions of the equation. $\\$$4 x-3^x=-6$

Determine whether each of the following is true or false. Assume that $a, x, M$, and $N$ are positive. $\\$$\log _a 8 x=\log _a x+\log _a 8$

Garlic Supply The amount of garlic in the food supply has significantly increased in the past $80 \mathrm{yr}$. The average U.S. per capita consumption of garlic has increased from about $0.15 \mathrm{lb}$ in the $1920 \mathrm{~s}$ to $2.5 \mathrm{lb}$ in the mid-2000s The following exponential function models this growth in the consumption of garlic:

$$G(x)=0.0882(1.0414)^x,$$

where $x$ is the number of years since 1925 . Estimate the amount of garlic consumed per capita in 1960 and in 1990.

Express in terms of sums and differences of logarithms.

$$\log \sqrt[3]{\frac{M^2}{N}}$$

Find each of the following using a calculator. Round to four decimal places. $\\$$\ln 50$

The amount of imports to the United States has increased exponentially since 1980. The exponential function

$$I(x)=297.539(1.075)^x,$$

where $x$ is the number of years after 1980 , can be used to estimate the total amount of U.S. imports, in billions of dollars. Find the total amount of imports to the United States in 1995, in 2005, in 2010, and in 2013. Round to the nearest billion dollars.

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 30$

Use a graphing calculator to find the approximate solutions of the equation. $\\$$x e^{3 x}-1=3$

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)= $$ \frac{5 x-3}{2 x+1}$ $$

Determine whether each of the following is true or false. Assume that $a, x, M$, and $N$ are positive. $\\$$\log _a x^3=3 \log _a x$

The amount and quality of milk produced increases when dairy cows are fed high-quality alfalfa. Although annual milk consumption per capita in China has been one of the lowest in the world, it has increased greatly in recent years. Consequently, the demand for imported alfalfa has increased exponentially. In 2008, approximately $20,000$ tons of U.S. alfalfa were imported by China. This number increased to approximately $650,000$ tons by 2013. The increase in imported U.S. alfalfa can be modeled by the exponential function

$$A(x)=22,611.008(1.992)^x,$$

where $x$ is the number of years after 2008. Find the number of tons of U.S. alfalfa imported by China in 2012 and in 2016.

The number of pages in the IRS Tax Code has increased from $400$ pages in 1913 to $70,320$ pages in 2009. The increase can be modeled by the exponential function

$$T(x)=400(1.055)^x,$$

where $x$ is the number of years since 1913. Use this function to estimate the number of pages in the tax code for 1950, 1990, and for 2000. Round to the nearest one.

Solve the given exercise below: $\\$$\log x=\ln x$

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)= $$ \frac{x+4}{x-3}$ $$