IB Math SL Year 2Name: ________________________Date: _________________________

Lesson 2-3: Properties of e, Inverse Exponential Functions, and Graphing Logs

In this lesson we will revisit the following learning goals:

1. What are the properties of exponential functions?2. How can we sketch the graph of an exponential function?3. What are the properties of e? What is the inverse of e?4. How do we use the binomial expansion?

Key Notes What do I need to know? Notes to Self1. Properties of e

2. Inverse Exponential Functions

Definitions for:o Exponential Functionso Logarithmic Functionso Asymptoteso x- and y-intercepts

Calculation for/Processes:o Equations for exponential and

logarithmic functionso Equations for horizontal and

vertical asymptoteso Graphing exponential functions and

log functions with transformations2. Binomial Expansion Definitions for:

o Expansiono Termo Power of a binomialo Binomial coefficient

Calculation for/Process:o Expanding a binomialo Calculating a binomial coefficiento Finding a specific term in a

binomial expansion

IB Math SL Year 2

Graphing Exponential and Logarithmic Functions This week we looked at graphs of exponential functions today I will refresh your memory.

- Remember that we can determine their horizontal asymptote by looking at the ___________up or down. -

o Ex: f (x)=2x This is not shifted vertically up or down so the horizontal asymptote is at ¿¿ .

oo Ex: f ( x )=2x−4This is shifted down 4, so the horizontal asymptote is at ¿¿ .

- When graphing exponential functions we should:-

o Determine the __________________________________ (see above)oo Determine the _________________ (plug in x=0)oo Determine a second point (maybe at x=1 or x=-1) oo Label these ___________ things and sketch a graph!

Example: Graph f ( x )=3x+2

Facts about Exponential Functions and their graphs:

- The domain is always all real numbers or (−∞,∞)

- The range is always restricted based off the horizontal asymptote (see above)

- The graph will always do one of two things-- increase (growth) or decrease (decay)

- On the right you can see the general exponential function y=ax, The y-intercept is (0,1)

IB Math SL Year 2Logarithmic Functions

*If we graphed BOTH an exponential function and it’s ______________, the logarithmic function of the same base we would see something like the graph here:

*Remember to graph an inverse function you reflect over the line y=x

In general, if f ( x )=ax, then f−1 (x )=loga x.*The inverse of an exponential function is a logarithmic function*

Graphs of Logarithmic Functions- Using our knowledge of exponential functions we can now graph logarithmic functions!

**“Inverse” what we know about exponential functions**

- We can determine their vertical asymptote by looking at the ________________ up or down. o Ex: y=loga x This is not shifted horizontally left or right vertical asymptote is at ¿¿

o Ex: y= loga(x−2) This is shifted right 2, so the vertical asymptote is at x=2. o Ex: y=loga(x¿+3)¿ This is shifted left 3 so the vertical asymptote is at ¿¿ .

Facts about Logarithmic Functions and their graphs: **“Inverse” what we know about exponential functions**

- The range is always all real numbers or (−∞,∞)

- The domain is always restricted based off the vertical asymptote(see above)

- The graph will always do one of two things-- increase or decrease

- On the right you can see the general exponential function y=loga x, The x-intercept is (1,0)

- STEPS for graphing logarithmic functions:

1. Determine the __________________ asymptote (see above)2. Determine the y-intercept (plug in x=0).

Careful! You can only take logs of _______ numbers so sometimes you won’t have one! 3. Determine the x-intercept (plug in y=0)

Determine a second point 4. Label these __________ things and sketch a graph!

IB Math SL Year 2

Let ’ s try some together:

1. Exam Style: Sketch the graph of y=−2 log ( x−1 )without using a calculator. If the graph crosses the axes determine the points where.

Now that we are done with this graph… did that seem tough?! It was a little! We will often have our calculator to graph log functions but if you don’t this is it!

Notice, the domain is (1 ,∞)

The range is (−∞,∞ )

IB Math SL Year 2What is e?

The numbers π and e have many similar properties. Both are irrational, meaning that they cannot be written as a fraction of two whole numbers.

Evaluate: (a) e2 (b) 3e7

What is ln?Ln is read as “________________________________________________” which has a base of __________.

How do we use it? With equations!

What is the difference between,

6 = 45x +2 and e1−3x = 15

Another way to use it…

Try one!

3 = ln(4x-2)

My thinking…

Swap the x and y.

Use properties that e ln “cancel”

IB Math SL Year 2

Your turn to practice!

1. Consider: y=log2 ( x+1 )−2 without a calculator!!a. What is the vertical asymptote of f(x)?

b. Determine one point on the curve by using your information of log22=1

c. What is the x-intercept? The y-intercept?

2. The sketch shows the graph of y=loga x. Find the value of a.

3. Given that f (x)=log3 x find f−1(x)

4.

IB Math SL Year 2

5.

6. If a = ln 2, and b = ln 5, write the following in terms of a an db(a) ln 50 (b) ln 0.16

7.  f is a function given by f (x) = log2(x + 2)

a. Find the domain of f and range of f.

b. Find the vertical asymptote of the graph of f.

c. Find the x and y intercepts of the graph of f if there are any.

d. Sketch the graph of f.