Advanced Algebra Trig Lesson Plans
Section 5.2 – Logarithmic Functions
Enduring Understandings: The student shall be able to:
1. Recognize and evaluate logarithmic functions
2. Graph logarithmic functions
3. Use logarithmic functions to solve real life problems
Standards:
10 Exponential and Logarithmic Functions
Defines, graphs, and shows the inverse relationship between logarithmic and
exponential functions
11 Exponential and Logarithmic Functions
Solves logarithmic and exponential equations and problems
12 Exponential and Logarithmic Functions
Solves problems involving application of exponential and logarithmic functions using
appropriate techniques and tools - Makes predictions from collected data using
regression techniques
Essential Questions:
 What is a logarithmic function, and how does it differ from an exponential
function?
 How are exponential and logarithmic functions related?
Activities:
We have been talking about exponential functions. We are now going to study
logarithmic functions.
Equivalent Exponential and Logarithmic forms: For any positive base b, where b  1,
bx = y iff x = logb y
Exponential form
103 = 1000
Logarithmic form:
3 = log10 1000
exponent
base (emphasize)
Fill in the following tables:
Complete for f(x) = 10x
x
-2
f(x)
-1
0
1
2
Complete the table for f(x) = log10x
x
1/100
1/10
f(x)
1
10
100
What do you notice? f(x) = 10x and f(x) = log10x are the inverse functions of each other.
The inverse of y = 10x is x = 10y, which in logarithmic form is y = log10 x.
What does inverse mean?
 Geometrically, they are reflected across the y = x line
 Algebraically, they “undo” each other, so f(g(x)) = g(f(x)) = x
A table can be used to find x for 10^x = 1000, but to find x for 10^x = 2.3 takes a
logarithm.
Write log as exponential # 2 – 6 even
Write exponential as log # 10 – 18 even
Properties of logarithms:
1. loga1 = 0 because a0 = 1
2. logaa = 1 because a1 = a
3. log a a x  x and a loga x  x inverse property
4. If logax = logay then x = y one=-to-one Property
The Natural Logarithmic Function is defined by: f(x) = logex = ln x, x > 0
The above Properties of Logarithms applies to natural logs, since e is just a specific base,
but they look different:
1. ln 1 = 0 because e0 = 1
2. ln e = 1 because e1 = e
3. ln e x  x and e ln x  x inverse property
4. If ln x = ln y then x = y one=-to-one Property
Evaluate expressions without a calculator # 20 – 26 even
Use a calculator to solve expressions # 34 – 40 even
Describe the graph if b = 1 in y = logb x. (the equivalent form is x = by, so if b = 1, we
would have x = 1 for an equation, i.e., a vertical line)
Is y = log1 x a function? (no – it does not pass the vertical line test)
Find the domain and x-intercept and vertical asymptote of log function: # 52 – 62 even
Applications:
Assessments: Hw pg 399 – 402, # 1 – 61 by 4’s, 75, 77, 79, 83, 85 (21)