Test #3 ReviewTest will cover Modules 12, 13, 14, and 15

Remember that logs of numbers are still just numbers. Please don’t turn them into decimals unless instructed otherwise, it’s like leaving a square root as a square root – it’s just prettier!

Do not be afraid of e. It’s just a number too. It just happens to be a super cool number that we can do a lot with.

This rule will be your friend:

Need to know facts:

Remember:

Exponential Functions:

Log Functions:

Forms:

where a > 0 and a ≠ 1.

where a > 0 and a ≠ 1.

You don’t have a chance at doing graphing transformations correctly if you don’t start with the correct parent function. Remember the 4 basic exponential/logarithmic shapes:◦ If you forget, you can always plug in a couple of points to help

you remember which one is which (you can even do this to check that you’ve done a transformation correctly!)

Need to know facts (graphing):

𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1 𝑓 (𝑥 )=𝑎𝑥 ,𝑎>1

𝑓 (𝑥 )=log𝑎 𝑥 ,0<𝑎<1 𝑓 (𝑥 )=log𝑎 𝑥 ,𝑎>1

Need to know facts (graphing): Also, remember the vertical and horizontal asymptotes

of the parent functions to make it easier to see the asymptotes in the transformed ones.

𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1 𝑓 (𝑥 )=𝑎𝑥 ,𝑎>1

𝑓 (𝑥 )=log𝑎 𝑥 ,0<𝑎<1 𝑓 (𝑥 )=log𝑎 𝑥 ,𝑎>1

𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1 𝑓 (𝑥 )=𝑎𝑥 ,𝑎>1

𝑓 (𝑥 )=log𝑎 𝑥 ,0<𝑎<1 𝑓 (𝑥 )=log𝑎 𝑥 ,𝑎>1

Domain: all realsRange: (0, infinity)

Domain: all realsRange: (0, infinity)

Domain: (0, infinity)Range: all reals

Domain: (0, infinity)Range: all reals

Graph Adjustments Vertical Adjustments◦ f(x) + c

Moves graph up c units◦ f(x) – c

Moves graph down c units◦ 2*f(x)

Stretches vertically by a factor of 2 (could be any number > 1)

◦ 0.5*f(x) Compresses vertically by a factor of 2

(any fraction between 0 and 1)◦ -f(x)

Reflection over the x axis

Graph Adjustments Horizontal Adjustments

(usually backwards from what you expect)◦ f(x + c)

Moves graph left c units◦ f(x – c)

Moves graph right c units◦ f(2*x)

Compresses horizontally by a factor of (1/2) (could be any number > 1)

◦ f(0.5*x) Stretches by a factor of 2

(any fraction between 0 and 1)◦ f(-x)

Reflection over the y axis

Log Properties: Identity:

Inverse (I): ◦

Inverse (II):

Exponent to Constant:

Product:

Quotient:

Also…remember that ◦!◦ does not exist! And neither does the log of any

negative number because these values are not in the domain of the log!

Log Properties