How are the functions \(f\) and \(g\) given by \(f(x) = (1.05)^x\) and \(g(x) = e^{0.05x}\) similar? How are they different?

1. An investment is worth \$100 and grows in value by 4 percent each year. Explain why the value of the investment after \(t\) years is given by \(100 \boldcdot 1.04^t\).
2. A second investment is worth \$100 and grows in value by 2 percent each half year. Explain why the value of the investment after \(t\) years is given by \(100 \boldcdot (1.02)^{2t}\).
3. A third investment is worth \$100 and grows in value by 4 percent per year, but the interest is applied continuously, at every moment. The value of this investment after \(t\) years is given by \(100 \boldcdot e^{(0.04t)}\). Order the investments from slowest growing to fastest growing. Explain how you know.

The population of a country is growing exponentially, doubling every 50 years. What is the annual growth rate? Explain or show your reasoning.

If \(b>0\), what is the value of \(\log_b \left(\frac{1}{b} \right)\)? Explain or show your reasoning.

The expression \(5 \boldcdot \left(\frac12\right)^d\) models the amount of a radioactive substance, in nanograms, in a sample over time in decades, \(d\). (1 nanogram is a billionth, or \(1 \times 10^{\text-9}\), of a gram.)

1. What do the 5 and the \(\frac12\) tell us in this situation?
2. When will the sample have less than 0.5 nanogram of the radioactive substance? Express your answer to the nearest half decade. Show your reasoning.
3. Show that only about 5 picograms of the substance will remain one century after the sample is measured. (A picogram is a trillionth, or \(1 \times 10^{\text-12}\), of a gram.)

Select all true statements about the number \(e\).