Logarithmic Functions in Algebra 2 Honors Worksheet
Reminder for Graded Assessment
The graded assessment for sections 6.1-6.3 is scheduled for Monday. Today's lesson will be supplemental.
Class Activity
For the Do Now activity:
- Check the answer key for the "Practice with logs" activities and ask any questions.
- Open the "6.3 Day 1 Notes" from the last class and go to the "Exploration" section.
- Identify the pattern noticed from the chart.
Given:
Rewrite in base 10
- log10
- log₁0
- 10¹
- log1000
- log10
- log1,000,000
- log 10,000,000,000
- 10³
- log₁00
- log₁0 11
Use the observed pattern to simplify the following:
- log₂27 = -7
- log₅65
- log₈(52)x
- log₆(12x)
Exponential and Logarithmic Functions
By the definition of a logarithm, it follows that the logarithmic function g(x)=log, x
is the inverse of the exponential function f(x)=b*. This means that
g(f(x)) = log bª
and
f(g(x)) = _b%x
In other words, exponential functions and logarithmic functions "undo" each other.
Practice
Simplify each expression:
- log₃27
- Evaluate 1/3 * 6^>10
- log₅92
- log₆(322)
- f(x) = g(x) = x
- Ine²x = 2x
- loge e
Example 1
Find the inverse of each function:
- f(x)=24*-3: Y=-3
- f"(x) = logu (x + 3) = -√ A² X=-3
Example 2 (You Try)
- f(x)=ln(x-5)
- y = e*+s
Graphing Logarithmic Functions
f(x) = log₂ x+3
- x=10924434
- x-3=109₂4
- 2x-3=
- f"(x) = 2*-s
Let's Talk Graphs!
We know f(x)=b* and g(x)= log₁ x are inverses.
Asymptote: X=0
Domain:(,)
Range: ∞
Core Concept: Parent Graphs for Logarithmic Functions
The graph of f(x) = log, x is shown below for b> and for 0 < b < 1. Because
f(x) = log,x and g(x) = b are inverse functions, the graph of f(x) = log, x is the
reflection of the graph of g(x)= b in the line y = x.
Graph of f(x) = log, x for b> 1
Graph of f(x) = log₁ x for 0 < b < 1
Note that the y-axis is a vertical asymptote of the graph of f(x) = log, x. The domain of f(x) = log, xis x > 0. and the range is all real numbers.
Given y = alog(x-4) +k ⇒ Asymposte at x=h
Graphing Strategy
- Rewrite the function in exponential form.
- Create a table of values.
- Substitute y-values.
Example 3 (You Try): Graph f(x)= -log₁x-1
Asymptote: x=0
Domain: (0,00)
Range: (-00,00)
