# Lesson 16

Using Graphs and Logarithms to Solve Problems (Part 2)

### Problem 1

The revenues of two companies can be modeled with exponential functions $$f$$ and $$g$$. Here are the graphs of the two functions. In each function, the revenue is in thousands of dollars and time, $$t$$, is measured in years. The $$y$$-coordinate of the intersection is 215.7. Select all statements that correctly describe what the two graphs reveal about the revenues.

A:

The intersection of the graphs tells us when the revenues of the two companies grow by the same factor.

B:

The intersection tells us when the two companies have the same revenue.

C:

At the intersection, $$f(t) >g(t)$$.

D:

At the intersection, $$f(t) = 215.7$$ and $$g(t)=215.7$$.

E:

We need to know both expressions that define $$f$$ and $$g$$ to find the value of $$t$$ at the intersection.

F:

If we know at least one of the expressions that define $$f$$ and $$g$$, we can calculate the value of $$t$$ at the intersection.

### Problem 2

The population of a fast-growing city in Texas can be modeled with the equation $$p(t) = 82 \boldcdot e^{(0.078t)}$$. The population of a fast-growing city in Tennessee can be modeled with $$q(t) = 132 \boldcdot e^{(0.047t)}$$. In both equations, $$t$$ represents years since 2016 and the population is measured in thousands. The graphs representing the two functions are shown. The point where the two graphs intersect has a $$y$$-coordinate of about 271.7.

1. What does the intersection mean in this situation?
2. Find the $$x$$-coordinate of the intersection point by solving each equation. Show your reasoning.
1. $$p(t) = 271.7$$
2. $$q(t)=271.7$$
3. Explain why we can find out the $$t$$ value at the intersection of the two graphs by solving $$p(t) = q(t)$$.

### Problem 3

The function $$f$$ is given by $$f(x) = 100 \boldcdot 3^x$$. Select all equations whose graph meets the graph of $$f$$ for a positive value of $$x$$.

A:

$$y = 10 \boldcdot e^x$$

B:

$$y = 500 \boldcdot e^x$$

C:

$$y = 500 \boldcdot e^{\text-x}$$

D:

$$y = 1,\!000 \boldcdot 2^x$$

E:

$$y = 600 \boldcdot 10^x$$

### Problem 4

The half-life of nickel-63 is 100 years. A students says, “An artifact with nickel-63 in it will lose a quarter of that substance in 50 years.”

Do you agree with this statement? Explain your reasoning.

### Solution

(From Unit 4, Lesson 7.)

### Problem 5

Technology required. Estimate the value of each expression and record it. Then, use a calculator to find its value and record it.

expression estimate calculator value
$$\log 123$$
$$\log 110,\!000$$
$$\log 1.1$$

### Solution

(From Unit 4, Lesson 11.)

### Problem 6

Here are graphs of the functions $$f$$ and $$g$$ given by $$f(x) = 100 \boldcdot (1.2)^x$$ and $$g(x) = 100 \boldcdot e^{0.2x}$$.

Which graph corresponds to each function? Explain how you know.

### Solution

(From Unit 4, Lesson 13.)

### Problem 7

Here is a graph that represents $$f(x) = e^x$$

Explain how we can use the graph to estimate:

1. The solution to an equation such as $$300 = e^x$$.
2. The value of $$\ln 700$$.