3.2 Logarithmic Functions and Their Graphs

Understanding Logarithmic Functions

Definition of a logarithmic function: the inverse of an exponential function where a is the base

where the anti exponential function is equal to the logarithmic function

Note: Exponential function have x-inputs, while the logarithmic functions have x-outputs

Evaluating Logarithmic Functions:

Example #1: 

Write the logarithmic equation in exponential form 

 

Solution: 2 is the base of the equation, or a. Therefore according to the definition of a logarithmic function, the exponential form is:

where y=8, x=3, and a=2

Example #2:

Solve the equation for x

Solution: put the logarithmic equation in exponential form 

then simplify the equation so that you have a common base

in order to solve for x, set the exponents equal to one another, such as:

and solve for x

Final Answer: 

Example #3

Solve the equation for x

Solution: put the equation in exponential form

Final Answer: Impossible, because there is not an exponent that makes 3 equal -81

Calculator Tips

There are two functions that are used so commonly, they have made functions on the calculator making solving the equations faster and more efficient

Common log- 

Natural log-

Graph of a log function

Remember that a log function is the inverse of an exponential function, so it's the graph of a logarithmic equation is a reflection of the exponential equation over the line y=x

the red line graph represents the log function, as you can see it is a reflection over the line y=x

Exponential Function Properties:

D: 

R:

x-intercept: none 

y-intercept: (0,1)

Vertical Asymptote: none

Horizontal Asymptote: y=0

Logarithmic Function Properties:

D:

R:

x-intercept: (1,0)

y-intercept: none

Vertical Asymptote: x=0

Horizontal Asymptote: none

*NOTE: the domain and range, as well as the intercepts and asymptotes, are inverses of the exponential function(they switch)

Shifting of a logarithmic graph

d=shift vertical(upwards and downwards)

c=shift horizontally(right and left)

a=vertical stretch/compression, if (-) it is reflected over the x-axis

b= when b is larger, it causes the incline to happen slower, rate of change