# Lesson 14

Solving Exponential Equations

### Problem 1

Solve each equation without using a calculator. Some solutions will need to be expressed using log notation.

1. $$4 \boldcdot 10^x = 400,\!000$$
2. $$10^{(n+1)} = 1$$
3. $$10^{3n} = 1,\!000,\!000$$
4. $$10^p = 725$$
5. $$6 \boldcdot 10^t = 360$$

### Problem 2

Solve $$\frac14 \boldcdot 10^{(d+2)} = 0.25$$. Show your reasoning.

### Problem 3

Write two equations—one in logarithmic form and one in exponential form—that represent the statement: “the natural logarithm of 10 is $$y$$”.

### Problem 4

Explain why $$\ln 1 = 0$$.

### Problem 5

If $$\log_{10}(x) = 6$$, what is the value of $$x$$? Explain how you know.

### Solution

(From Unit 4, Lesson 9.)

### Problem 6

For each logarithmic equation, write an equivalent equation in exponential form.

1. $$\log_2 16 = 4$$
2. $$\log_3 9 = 2$$
3. $$\log_5 5 = 1$$
4. $$\log_{10} 20 = y$$
5. $$\log_2 30 = y$$

### Solution

(From Unit 4, Lesson 10.)

### Problem 7

The function $$f$$ is given by $$f(x) = e^{0.07x}$$.

1. What is the continuous growth rate of $$f$$?
2. By what factor does $$f$$ grow when the input $$x$$ increases by 1?