Today's soundtrack is Grits: Dichotomy A, a rap album that I bought because I'd heard the group's song "Ooh Ahh" on a sampler CD and thought it was awesome.
This evening, I'm expanding on what I learned yesterday about graphing exponential functions; now, I'll be learning how to graph logarithmic functions.
The inverse of an exponential function is a logarithmic function. Inverse functions are reflected on the x=y line, which runs diagonally, rising from left to right.
The form of an inverse exponential looks like this:
Inverse Function 1
Exponential: y=5ˣ
Logarithmic: y=logₓ5
Inverse Function 2
Exponential: y=0.5ˣ
Logarithmic: y=log₀.₅x
Inverse Function 3
Logarithmic: y=log₃x
Exponential: y=3ˣ
How to graph a logarithmic function
Write the basic function in exponential form
If we start with y=log₃x, we "bump the log" by pulling out the log's base, then we make it the coefficient on the other side of the equation, making the other variable its exponent
y=log₃x becomes 3ʸ=x
Create a table of value
Fill the table with appropriate values of y
Find the corresponding x-values algebraically
Determine and apply any transformations (translation, expansion/compression, reflection)
Plot the graph
Draw a smooth line connecting the dots
How to Find the Properties of a Logarithmic Function
Write the function in exponential form
y-intercept
Substitute 0 for x and solve algebraically.
A domain error means that there is no y-int
x-intercept
Substitute 0 for y and solve algebraically
A domain error means that there is no x-int
Domain and range:
We can't log 0 or negative numbers; any variable that is the argument of the log must therefore be greater than 0.
Any variable that is not attached to a log can be any real number (x∈ℝ or y∈ℝ).
Asymptote
Since the log cannot equal 1 or be less than 0, solve x and its related numbers inside the brackets, if applicable
A quick tip: When we graph functions of the form y=logₐx, if a is greater than 0, the graph will start at the bottom, then curve up and to the right. If a is less than 0, the graph will start at the top, then go down as it goes right. Also, I've mentioned this before, but it bears repeating: If the base is greater than 1, the higher the value, the sharper the curve; if the base is a rational number less than 1, the lower the value, the sharper the curve will be.
That concludes this unit! Over the next couple of days, I'll be doing review of polynomials, exponents, and logarithms.