# Lesson 9

What is a Logarithm?

- Let’s learn about logarithms.

### 9.1: Math Talk: Finding Solutions

Find or estimate the value of each variable mentally.

\(4^a=16\)

\(4^b=2\)

\(4^\frac{5}{2}=c\)

\(4^d=56\)

### 9.2: A Table of Numbers

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

2 | 0.3010 |

3 | 0.4771 |

4 | 0.6021 |

5 | 0.6990 |

6 | 0.7782 |

7 | 0.8451 |

8 | 0.9031 |

9 | 0.9542 |

10 | 1 |

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

20 | 1.3010 |

30 | 1.4771 |

40 | 1.6021 |

50 | 1.6990 |

60 | 1.7782 |

70 | 1.8451 |

80 | 1.9031 |

90 | 1.9542 |

100 | 2 |

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

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 |

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

2,000 | 3.3010 |

3,000 | 3.4771 |

4,000 | 3.6021 |

5,000 | 3.6990 |

6,000 | 3.7782 |

7,000 | 3.8451 |

8,000 | 3.9031 |

9,000 | 3.9542 |

10,000 | 4 |

- Analyze the table and discuss with a partner what you think the table tells us.
- Use the table to find the value of the unknown exponent that makes each equation true.
- \(10^w = 1,\!000\)
- \(10^y = 9\)
- \(10^z = 90\)

- Notice that some values in the columns labeled \(\log_{10} x\) are whole numbers, but most are decimals. Why do you think that is?

### 9.3: Hello, Logarithm!

- Here are two true equations based on the information from the table:
\(\begin {align} \log_{10} 100 &= 2\\ \log_{10} 1,\!000 &= 3 \end{align}\)

What values could replace the “?” in these equations to make them true?

- \(\log_{10} 1,\!000,\!000 = {?}\)
- \(\log_{10} 1 = {?}\)
- \(\log_{10} ? = 4\)

- Between which two whole numbers is the value of \(\log_{10} 600\)? Be prepared to explain how do you know.
- The term
*log*is short for**logarithm**. Discuss the following questions with a partner and record your responses:- What do you think logarithm means or does?
- Next to “log” is a subscript—a number or letter printed smaller and below the line of text. What do you think the subscript tells us?
- What about the other two numbers on either side of the equal sign (for example, the 100 and the 2 in \(\log_{10} 100 = 2\))? What do they tell us?

- For which whole number values of \(n\) is \(\log_{10}(n)\) an integer?
- Why will \(\log_{10}(n)\) never be equal to a non-integer rational number?

### Summary

We know how to solve equations such as \(10^a = 10,\!000\) or \(10^b= \frac{1}{100}\) by thinking about integer powers of 10. The solutions are \(a = 4\) and \(b = {\text-}2\). What about an equation such as \(10^p = 250\)?

Because \(10^2 = 100\) and \(10^3 = 1,\!000\), we know that \(p\) is between 2 and 3. We can use a **logarithm** to represent the exact solution to this equation and write it as:

\(\displaystyle p=\log_{10} 250\)

The expression is read “the log, base 10, of 250.”

- The small, slightly lowered “10” refers to the base of 10.
- The 250 is the value of the power of 10.
- \(\log_{10} 250\) is the value of the exponent \(p\) that makes \(10^p\) equal 250.

Base 10 logarithms are often written without the number 10. So \(\log_{10} 250\) can also be written as \(\log 250\) and this expression is read “the log of 250.”

One way to estimate logarithms is with a logarithm table. For example, using this base 10 logarithm table we can see that \(\log_{10} 250\) is between 2.3 and 2.48.

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

2 | 0.3010 |

3 | 0.4771 |

4 | 0.6021 |

5 | 0.6990 |

6 | 0.7782 |

7 | 0.8451 |

8 | 0.9031 |

9 | 0.9542 |

10 | 1 |

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

20 | 1.3010 |

30 | 1.4771 |

40 | 1.6021 |

50 | 1.6990 |

60 | 1.7782 |

70 | 1.8451 |

80 | 1.9031 |

90 | 1.9542 |

100 | 2 |

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

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 |

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

2,000 | 3.3010 |

3,000 | 3.4771 |

4,000 | 3.6021 |

5,000 | 3.6990 |

6,000 | 3.7782 |

7,000 | 3.8451 |

8,000 | 3.9031 |

9,000 | 3.9542 |

10,000 | 4 |

### Glossary Entries

**logarithm**The logarithm to base 10 of a number \(x\), written \(\log_{10}(x)\), is the exponent you raise 10 to get \(x\), so it is the number \(y\) that makes the equation \(10^y = x\) true. Logarithms to other bases are defined the same way with 10 replaced by the base, e.g. \(\log_2(x)\) is the number \(y\) that makes the equation \(2^y = x\) true. The logarithm to the base \(e\) is called the natural logarithm, and is written \(\ln(x)\).