Kelly's math stuff
Review Logarithmic Functions
|Logarithmic Functions: A name for the inverse of an exponential function |
|Given the function [pic], the inverse function is [pic] |
|which is read, the logarithm, base 2, of x. |
|[pic] means "y" is the exponent that "a" must be raised to, in order | [pic] |
|to get the value x. | |
|[pic] means "x" is the power that 2 must be raised to, in order to | [pic] |
|get 8. |Therefore, x = 3. |
|Converting from Logarithmic to Exponential Functions |
|1. Remember that x > 0 and "a" is a positive constant other than 1. |
|2. The answer to a logarithmic equation is always an exponent. |
| |
|The answer to a log is the exponent |
| |
|[pic] is equivalent to [pic] |
| |
|that the base is raised to in order to make "x" |
|Logarithmic Equations Converted to Exponential Equations |
|[pic] |[pic] |
| |x = 8 |
|[pic] |[pic] |
| |x = 5 |
|[pic] | |
|[pic] |[pic] |
| |[pic] |
| |x = 2 |
|[pic] |[pic] |
|[pic] |[pic] |
|Properties of Logarithmic Functions: Since logarithms are exponents, they |
|will follow the same rules for exponents. |
|In multiplication you add exponents|[pic] |
| |the log of a product equals the sum of the logs. |
| |[pic] |
| |Example: [pic] |
|In division you subtract exponents |[pic] |
| |the log of a quotient equals the difference of the logs |
| |[pic] |
| |Example: [pic] |
|In power to power you multiply |[pic] |
|exponents |The log of a power equals the power times the log of the number. |
| |[pic] |
| |Example: [pic] |
| | |
F. Natural Logarithms: ln Same rules as for logs, but uses e (2.7182818284…) as the base
instead of 10.
Example:
[pic]
Example:
[pic] WHY??
G. Solving Exponential and Logarithmic Equations: ax = ay . x = y
Examples: a) 3x = 81 b) 43x+5 = 16
3x = 34 43x+5 = 42
x = 4 3x + 5 = 2
x = -1
c) Solve: log (3x - 10) - log 5x = -1
[pic] rewrite as a single logarithm
[pic] rewrite in exponential form
[pic] simplify the negative exponent and solve as a normal linear equation
5x = 30x - 100
-25x = -100
x = 4
d) Solve: e-0.04t = 0.08
ln e-0.04t = ln 0.08 Take ln of both sides
(-0.04t) ln e = ln 0.08 Use exponent rule for a logarithm, ln e = 1,
-0.04t = ln 0.08 Enter in the calculator to
determine a value for the logarithm.
-0.04t = -2.5257286 Solve for t
t = 63.14
H. Change of Base Formula
[pic] Example: [pic]
Derivatives of Logs and Exponentials
For b > 0 and b ≠ 1
[pic] Example. [pic]
[pic] Proof: [pic]
[pic] Example. [pic]
[pic] (x > 0) Proof: [pic]
Sample Problems
Find [pic]
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
Self-Check 10
Find [pic]
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
Answers:
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
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