# Lesson 14

Solving Exponential Equations

- Let’s solve equations using logarithms.

### Problem 1

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

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

### 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.

### Problem 6

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

- \(\log_2 16 = 4\)
- \(\log_3 9 = 2\)
- \(\log_5 5 = 1\)
- \(\log_{10} 20 = y\)
- \(\log_2 30 = y\)

### Problem 7

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

- What is the continuous growth rate of \(f\)?
- By what factor does \(f\) grow when the input \(x\) increases by 1?