Warm-up: Simplify



Warm-up: Simplify.

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

Notes about Logarithms

1. What if you cannot make the bases the same? Consider: [pic].

How do you solve an equation where the unknown variable is in the exponent? ______________________________.

2. A logarithm is an _____________________.

Evaluating Logarithms

a) [pic] b) [pic] c) [pic] d) [pic]

e) [pic] f) [pic] g) [pic] h) [pic]

i) [pic] j) [pic] k) [pic] l) [pic]

Properties of Logarithms

1. [pic] = _____ 2. [pic] = _____ 3. [pic] = _____ 4. [pic] = _____

Change of Base Formulas

Example 1: Evaluate [pic]

|Base |Base 10 |Base e |

|Formula |[pic] |[pic] |

| | | |

| | | |

Example 2: Evaluate [pic] using

a) the change of base formula with common logs (round to 4 decimal places).

b) the change of base formula with natural logs (round to 4 decimal places).

Properties of Logarithms

| |Logarithm with Base a |Natural Logarithm |

|Product Property |[pic] |[pic] |

|Quotient Property |[pic] |[pic] |

|Power Property |[pic] |[pic] |

Example 3: Write each logarithm in terms on ln 2 and ln 3.

a) ln 6 b) ln [pic]

Practice Problem 2: Write each logarithm in terms of ln 2 and ln 5.

a) ln 10 b) ln [pic]

Example 4: Use the properties of logs to expand each expression.

a) [pic] b) [pic]

Practice Problem 3: Use the properties of logs to expand each expression.

a) [pic] b) [pic]

Example 5: Use the properties of logs to condense each expression to a single log (or ln).

a) [pic] b) [pic]

c) [pic]

Example 6

If [pic] and [pic], express the following in terms of a and b.

a) [pic] b) [pic] c) [pic]

Solving Logarithmic Equations

1. Logarithmic and Exponentials Functions are _______________.

2. Solve log equations (undo the log) by _____________________________________________________________.

3. Solve exponential equations (undo the exponent) by _________________________________________________.

Examples of Solving Log and Exponential Equations

1. [pic] 2. [pic] 3. [pic] 4. [pic]

Class Work

Evaluate the logarithm using the change of base formula. Round to three decimals.

1. [pic] 2. [pic]

Evaluate the logarithm using the properties of logs, given

[pic] [pic] [pic]

3. [pic] 4. [pic]

Use the properties of logs to expand the expression.

5. [pic] 6. [pic]

Use the properties of logs to condense the expression to a single log or ln.

7. [pic] 8. [pic]

Use the properties of logs to rewrite the expression in terms of r, s, and t given

[pic] [pic] [pic]

9. [pic] 10. [pic]

Solve for x.

11. [pic] 12. [pic]

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