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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