# Lesson 10

Interpreting and Writing Logarithmic Equations

- Let’s look at logarithms with different bases.

### 10.1: Reading Logs

The expression \(\log_{10} 1,\!000 = 3\) can be read as: “The log, base 10, of 1,000 is 3.”

It can be interpreted as: “The exponent to which we raise a base 10 to get 1,000 is 3.”

Take turns with a partner reading each equation out loud. Then, interpret what they mean.

- \(\log_{10} 100,\!000,\!000 = 8\)
- \(\log_{10} 1 = 0\)
- \(\log_2 16 = 4\)
- \(\log_5 25 = 2\)

### 10.2: Base 2 Logarithms

\(x\) | \(\log_2 (x)\) |
---|---|

1 | 0 |

2 | 1 |

3 | 1.5850 |

4 | 2 |

5 | 2.3219 |

6 | 2.5850 |

7 | 2.8074 |

8 | 3 |

9 | 3.1699 |

10 | 3.3219 |

\(x\) | \(\log_2 (x)\) |
---|---|

11 | 3.4594 |

12 | 3.5845 |

13 | 3.7004 |

14 | 3.8074 |

15 | 3.9069 |

16 | 4 |

17 | 4.0875 |

18 | 4.1699 |

19 | 4.2479 |

20 | 4.3219 |

\(x\) | \(\log_2 (x)\) |
---|---|

21 | 4.3923 |

22 | 4.4594 |

23 | 4.5236 |

24 | 4.5850 |

25 | 4.6439 |

26 | 4.7004 |

27 | 4.7549 |

28 | 4.8074 |

29 | 4.8580 |

30 | 4.9069 |

\(x\) | \(\log_2 (x)\) |
---|---|

31 | 4.9542 |

32 | 5 |

33 | 5.0444 |

34 | 5.0875 |

35 | 5.1293 |

36 | 5.1699 |

37 | 5.2095 |

38 | 5.2479 |

39 | 5.2854 |

40 | 5.3219 |

- Use the table to find the exact or approximate value of each expression. Then, explain to a partner what each expression and its approximated value means.
- \(\log_2 2\)
- \(\log_2 32\)
- \(\log_2 15\)
- \(\log_2 40\)

- Solve each equation. Write the solution as a logarithmic expression.
- \(2^y = 5\)
- \(2^y=70\)
- \(2^y=999\)

### 10.3: Exponential and Logarithmic Forms

These equations express the same relationship between 2, 16, and 4:

\(\displaystyle \log_2 16 = 4\)

\(\displaystyle 2^4 = 16\)

- Each row shows two equations that express the same relationship. Complete the table.
exponential form logarithmic form a. \(2^1= 2\) b. \(10^0=1\) c. \(\log_3 81 = 4\) d. \(\log_5 1 =0\) e. \(10^\text{-1}=\frac{1}{10}\) f. \(9^\frac12=3\) g. \(\log_2 \frac18= \text - 3\) h. \(2^y=15\) i. \(\log_5 40 = y\) j. \(b^y = x\) - Write two equations—one in exponential form and one in logarithmic form—to represent each question. Use “?” for the unknown value.
- “To what exponent do we raise the number 4 to get 64?”
- “What is the log, base 2, of 128?”

- Is \(\log_{2}(10)\) greater than 3 or less than 3? Is \(\log_{10}(2)\) greater than or less than 1? Explain your reasoning.
- How are these two quantities related?

### Summary

Many relationships that can be expressed with an exponent can also be expressed with a logarithm. Let’s look at this equation: \(\displaystyle 2^7 = 128\) The base is 2 and the exponent is 7, so it can be expressed as a logarithm with base 2:

\(\log_2 128 = 7\)

In general, an exponential equation and a logarithmic equation are related as shown here:

Exponents can be negative, so a logarithm can have negative values. For example \(3^{\text-4} = \frac{1}{81}\), which means that \(\log_3 \frac{1}{81} = \text-4\).

An exponential equation cannot always be solved by observation. For example, \(2^x = 19\) does not have an obvious solution. The logarithm gives us a way to represent the solution to this equation: \(x=\log_2 19\). The expression \(\log_2 19\) is approximately 4.25, but \(\log_2 19\) is an exact solution.

### 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 \(\ln(x)\).