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### Course: Algebra (all content)>Unit 11

Lesson 30: Graphs of logarithmic functions (Algebra 2 level)

# Shape of a logarithmic parent graph

Sal graphs y=log₅(x). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• How do you graph logs with a base of, say, 3 on a TI-84(calculator)?
• There's a nice rule that log base n of X divided by log base n of Y is equal to log base Y of X, so log(27)/log(3) = log base 3 of 27 = 3.

Some TI calculators also have a logBASE(x,y) function that you can use for this purpose.
• what if the base is not given.. for example log 2122 or something... please explain..
• If the base is not shown, it is base 10. log wihtout a base means log base 10.
• so logs will never be proportional on a graph
• Good observation, indeed a logarithm is never a proportional function.
• Could you have a negative base so that you could have both negative and positive values for x?
• A negative base will actually just give you complex values, just like taking the logarithm of negative numbers.
• Ok the question i have is:

Sketch the graph of y=3-log2(x+2).

I understand that it basically says 3- log2^y = (x+2). But I dont understand how i would graph that.
• You want to start with some x's that don't make really weird y's. I started with x=2 then I got
y=3-log2(2+2)
y=3-log2(4)
y=3-2
y=1
so you end up with (2,1). I tried some other numbers and baisicly for x you want
x=(2^x)-2
Here are the points I got.
(-1.75,5), (-1.5,4), (-1,3), (0,2), (2,1), (6,0), (14,-1), (30,-3)
You won't get a line, but it is predictable.
If you start at (0,2) then you can just do (x/2,y+1) to get the points on QII and (2x,y-1) for QI and QIV. As far as I can figure x will never equal -2. I hope this made sense.
• In exponential functions, what is the dependent variable and in logarithmic functions? Since x is the independent variable in exponential functions and logarithmic function is the inverse, shouldn't the dependent variable be x in the inverse function? I am confused!
• In function notation, "x" merely expresses the input to the function. It doesn't bear any connection to the "x" used elsewhere in the problem, or in the definition of a different function.

If you named both the input and output variables, then you would necessarily need to swap them to make a valid statement. Thus if y = e^x then x = ln(y). More likely, however, you'll see people write f(x) = e^x and g(x) = ln(x) where f(x) and g(x) are inverse functions. That is formally expressed by the property that f(g(x)) = g(f(x)) = x. Note that "y" never enters into it!
• What would Log_1 (1) be equal to? Is that a zero or One?
• Logs cannot have a base of 0 or 1. So, that is not defined.
• In linear equations, isn't the y dependent on the x? Why is it that in logarithms, the x is dependent on the y?
• Because with exponents, y is dependent on x, and since a logarithm is the inverse of an exponent (it "undoes" an exponent), it is just a graph of an equation with an exponent (not necessarily an exponential equation) flipped by 90 degrees. Technically, y is dependent on x, but it is usually easier to solve for x at a given value of y, rather than the way we normally do.
• What happens when you graph a logarithmic function on a logarithmic scale?