Relationships of Exponential and Logarithmic Functions

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Relationships of Exponential and Logarithmic Functions by Mind Map: Relationships of Exponential and Logarithmic Functions

1. 3-2 Logarithmic Functions

1.1. Range is always (-infinity, infinity)

1.2. Definition: the inverse of f(x)= b^x (with base b)

1.3. Basic properties of logs

1.3.1. log(base b) b^y= y because, b^y =b^y

1.3.2. Common log: a logarithm with base 10 or log(base 10)

1.3.2.1. y= log x if 10^y = x (x>0)

1.4. Evaluate Logarithms

1.4.1. A logarithm is an exponent

1.5. Natural Logarithms

1.5.1. Alog with base e or log(base e)

1.5.2. y = ln x , e^y =x, (x>0)

2. 3-1 Exponential Functions

2.1. Domain is always (-infinity, infinity)

2.2. Definition: The base is a constant and the exponent is a variable

2.2.1. 3^x NOT x^3

2.3. Graphing Transformations

2.3.1. Same techniques used to transform graphs of algebraic functions

2.3.1.1. Horizontal Shift

2.3.1.2. Dilations (horizontal and vertical)

2.3.1.3. Reflections (horizontal and vertical

2.3.1.4. Vertical Shift

2.4. Natural Base Exponential Function

2.4.1. Most real-world problems with exponential functions use base e

2.4.1.1. called natural base

2.4.1.2. irrational number

2.4.1.3. e=2.7183

2.5. Exponential Growth and Decay

2.5.1. Money

2.5.1.1. Compound Interest: A=P(1+r/n)^nt

2.5.1.2. Continuously Compounded Interest: A=Pe^rt

2.5.2. Population

2.5.2.1. Exponential Growth: N=Ne(1+r)^t

2.5.2.2. Continuous Exponential Growth: N=Noe^kt

3. 3-3 Properties of Logarithms

3.1. Simplifying & Expanding logs using properties

3.2. Product property

3.2.1. log(base b) xy = log(base b) x + log(base b) y

3.2.1.1. multiplication into addition

3.3. Quotient Property

3.3.1. log(base b) x/y = log(base b) x - log(base b) y

3.3.1.1. division into subtraction

3.4. Power Property

3.4.1. log(base b) x^p = plog(base b) x

3.4.1.1. exponent moved too in front of the log

3.5. Change of Base Formula

3.5.1. log(base b) x= log x/ log b

3.5.2. log(base b) x = ln x/ ln b

4. 3-4 Exponential & Logarithmic Equations

4.1. One-to-One property of Exponential Functions

4.1.1. One y-value is matched with one x-value

4.1.2. f(a)=f(b) then a=b

4.1.2.1. Can be used to solve simple exponential equations by getting a common base on each side of the equation

4.1.2.2. Can be used to solve logarithmic equations (exponentiating)

4.1.2.2.1. Effect of this is to convert the equation from logarithmic to exponential form

4.2. One-to-One property of Logarithmic Functions

4.2.1. log base b of x=log base b of y if y=x

4.2.1.1. Can be used to solve simple logarithmic equations

4.2.1.1.1. Condense so each side has same base

4.2.1.2. Can be used to solve exponential equations (taking the logarithm of each side)

4.2.1.2.1. Natural logs are more convenient, but and base can be used

4.3. Equations with multiple exponential expressions can be solved by factoring or the Quadratic Formula

4.3.1. Check for extraneous solutions

4.4. Equations with multiple logarithmic expressions can be solved by condensing

4.4.1. Then apply One-to-One property

4.4.1.1. Check for extraneous solutions

4.4.1.1.1. May not be obvious until checked with the original equation

5. 3-5 Modeling with Nonlinear Regression

5.1. Model data exhibiting exponential or logarithmic growth or decay

5.2. Exponential Regression

5.2.1. Real world problems

5.3. Logarithmic Regression

5.4. Analyaze Data & Linearize

5.4.1. Modeled by a quadratic function

5.4.2. An exponential function

5.4.3. A logarithmic function

5.4.4. A power function