1. Modules 12, 13, 14, 15 October 23, 2012

2.  Logs and exponentials are inverses of each other and can be rewritten in this way:  We can use the opposite function to isolate our variable when we solve equations.

3.  Exponential functions are of the form:  Our variable here is still x.  Ex.

4.  Log functions are of the form:  “What power do I raise the base a to in order to get the argument x?”  Ex.

5.  Exponential and log functions can also have transformations just like the functions did from the first exam material.  Ex.

7.  No, because here the base value would have to be a = -3, and we know that a has to be positive.

9.  Yes, because our base is a=3, which is valid. The negative out front is a reflection over the x-axis because it’s not being raised to the x power.

11.  Yes, because our base is a=2/3, which is valid because fractions are okay.

13.  Yes, because our base is a=π, which is valid because it is a value positive and not equal to 1. The 2 in front of the x is a horizontal transformation, which causes the graph to compress horizontally by ½.

15.  Yes, because our base is a=7, which is valid because it is a value positive and not equal to 1. The 4 can come out front by log rules, and so it will end up vertically stretching by 4.

17.  No, because our base is a = -2, and negative numbers aren’t allowed.

21.  Also think about the asymptotes to help you think about where they end up when you transform the graphs.  AND…  Our good old friends domain and range.

23. Domain: all reals Range: (0, infinity) Domain: all reals Range: (0, infinity) Domain: all reals Range: (0, infinity) Domain: all reals Range: (0, infinity) Domain: (0, infinity) Range: all reals Domain: (0, infinity) Range: all reals Domain: (0, infinity) Range: all reals Domain: (0, infinity) Range: all reals

25. New asymptote will be horizontal and at y=3 because it started at y=0 and there was a 3 unit vertical change. Domain: all reals Range: (- infinity, 3)

28.  When we have variables in the exponent, we need to take the log of both sides to “get it out” so we can solve for it.  Log rules

29.  We want to get x by itself, so we need to raise both sides to the x power (or reorganize using the definition).  Take the cube root of both sides.