# Lesson 9

What is a Logarithm?

- Let’s learn about logarithms.

### Problem 1

For each equation in the left column, find in the right column an exact or approximate value for the unknown exponent so that the equation is true.

### Problem 2

Here is a logarithmic expression: \(\log_{10}100\).

- How do we say the expression in words?
- Explain in your own words what the expression means.
- What is the value of this expression?

### Problem 3

The base 10 log table shows that the value of \(\log_{10} 50\) is about 1.69897. Explain or show why it makes sense that the value is between 1 and 2.

### Problem 4

Here is a table of some logarithm values.

- What is the approximate value of \(\log_{10}(400)\)?
- What is the value of \(\log_{10}(1000)\)? Is this value approximate or exact? Explain how you know.

\(x\) | \(\log_{10} (x)\) |
---|---|

200 | 2.3010 |

300 | 2.4771 |

400 | 2.6021 |

500 | 2.6990 |

600 | 2.7782 |

700 | 2.8451 |

800 | 2.9031 |

900 | 2.9542 |

1,000 | 3 |

### Problem 5

What is the value of \(\log_{10}(1,\!000,\!000,\!000)\)? Explain how you know.

### Problem 6

A bank account balance, in dollars, is modeled by the equation \(f(t) = 1,\!000 \boldcdot (1.08)^t\), where \(t\) is time measured in years.

About how many years will it take for the account balance to double? Explain or show how you know.

### Problem 7

The graph shows the number of milligrams of a chemical in the body, \(d\) days after it was first measured.

- Explain what the point \((1,2.5)\) means in this situation.
- Mark the point that represents the amount of medicine left in the body after 8 hours.

### Problem 8

The exponential function \(f\) takes the value 10 when \(x = 1\) and \(30\) when \(x = 2\).

- What is the \(y\)-intercept of \(f\)? Explain how you know.
- What is an equation defining \(f\)?