# Lesson 15

Using Graphs and Logarithms to Solve Problems (Part 1)

### Problem 1

- Use the graph to determine when the population will reach 100,000.
- Explain why \(\log_{2} 20\) also tells us when the population will reach 100,000.

### Problem 2

*Technology required.* Population growth in the U.S. between 1800 and 1850, in millions, can be represented by the function \(f\), defined by \(f(t) = 5 \boldcdot e^{(0.028t)}\).

- What was the U.S. population in 1800?
- Use graphing technology to graph the equations \(y=f(t)\) and \(y=20\). Adjust the graphing window to the following boundaries: \(0< x< 100\) and \(0< y< 40\).
- What is the point of intersection of the two graphs, and what does it mean in this situation?

### Problem 3

The growth of a bacteria population is modeled by the equation \(p(h) = 1,\!000 e^{(0.4h)}\). For each question, explain or show how you know.

- How long does it take for the population to double?
- How long does it take for the population to reach 1,000,000?

### Problem 4

What value of \(b\) makes each equation true?

- \(\log_b 144 = 2\)
- \(\log_b 64 = 2\)
- \(\log_b 64 = 3\)
- \(\log_b 64 = 6\)
- \(\log_b \frac19 = \text-2\)

### Problem 5

Put the following expressions in order, from least to greatest.

- \(\log_2 11\)
- \(\log_3 5\)
- \(\log_5 25\)
- \(\log_{10} 1,\!000\)
- \(\log_2 5\)

### Problem 6

Solve \(9 \boldcdot 10^{(0.2t)} = 900\). Show your reasoning.

### Problem 7

Explain why \(\ln 4\) is greater than 1 but is less than 2.