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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