Lesson 3
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FOR USE ONLY AS A GUIDE TO TAKE NOTES IN YOUR NOTEBOOK!
Unit e Ln 2: Exponential vs. Logarithmic expressions
Definition of Logarithmic Function:
For x > 0, a > 0, and a ≠1, [pic] if and only if [pic].
[pic] because [pic]
The function [pic] is called the logarithmic function with base a.
(Read as “log base a of x”)
CONVERTING BETWEEN LOGS AND EXPONENTIAL EXPRESSIONS:
Convert the following functions into either exponential or logarithmic form:
|Exponential equation |Logarithmic equation |
| |log3243 |
|10-1 = 0.1 | |
| |log64(4) |
|45 = 1024 | |
| |log8512 |
|65 = 7776 | |
Evaluating Logs with the Definition of Logarithm:
Logarithm with base 10 such as [pic] is called common logarithm and is written as [pic]
(base 10 is “invisible”) Common Logarithm can be evaluated on the calculator. NOTE: LOG key on your calculator indicates common log with base 10 only!
Examples: Evaluate the following functions using the definition of logs.
1. [pic] 2. [pic] 3. [pic] 4. [pic]
Examples: Evaluate the following expressions, using the definition of logs:
5. log5(25). 6. log2(8). 7. log64(4). 8. log6(6). 9. log4(–16). 10. log2(0).
Important Logarithmic Identities are:
[pic] because [pic]
[pic] because [pic]
[pic] because [pic]
[pic] [pic] there is no way we can raise a positive base to some power
and end up with zero.
[pic] [pic] ( x = y
Evaluating Logs that do not simplify easily
There is a formula that you can use with logarithms that do not simplify easily. For instance, try to evaluate [pic].
Change of Base Formula: For any positive real numbers [pic]
This formula puts both logs into the base of 10, which your calculator can help you evaluate.
Examples: Evaluate the following expressions, using the change of base formula.
a. [pic] b. [pic] c. [pic]
Mixed Practice: Evaluate the following expressions/equations, using the definitions and properties of logs.
1. [pic] 2. [pic] 3. [pic] 4. [pic]
5. [pic] 6. [pic] 7. [pic] 8. [pic]
9. [pic] 10. [pic]=x 11. [pic] 12. [pic]
13. [pic] 14. [pic] 15. [pic] 16. [pic]
17. [pic] 18. [pic] 19. [pic] 20. [pic]
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