, where x is the number of egg masses in thousands. If the percent of defoliation is 20%, approximately how many egg masses are there?
Lesson 22: Graphing Logarithmic Functions Lesson 22: Graphing Logarithmic Functions
Learning Outcome(s)
Learning Outcome(s): At the end of the lesson, the learner is able to represent a logarithmic function through its table of values, graph, and equation, find the domain and range of a logarithmic function, and graph logarithmic functions
Lesson Outline:
Lesson Outline:
1. Graph of y = logbx for b > 1 and for 0 < b < 1
2. Domain, range, intercepts, zeroes, and asymptotes of logarithmic functions 3. Graphs of transformations of logarithmic functions
10Schulz, K. (2015). The really big one. http://www.newyorker.com/magazine/2015/07/20/the-really-big-one
11Stewart, J., Redlin, L., & Watson, S. (2012). Precalculus: Mathematics for calculus (6th ed). Belmont, CA: Brooks/Cole, Cengage Learning.
D E P E D C O P Y
In the following examples, the graph is obtained by first plotting a few points. Results will be generalized later on.
Example 1.
Example 1. Sketch the graph of y = log2x.
Solution.
Step 1: Construct a table of values of ordered pairs for the given function. A table of values for y = log2x is as follows:
Step 2: Plot the points found in the table, and connect them using a smooth curve.
Plotting of points forof y = log2x Graph ofy = log2x
It can be observed that the function is defined only for x > 0. The function is strictly increasing, and attains all real values. As x approaches 0 from the right, the function decreases without bound, i.e., the line x = 0 is a vertical asymptote.
Example 2.
Example 2. Sketch the graph of y = log1/2x.
Solution.
Step 1: Construct a table of values of ordered pairs for the given function. A table of values for y = log2x is as follows:
D E P E D C O P Y
Step 2: Plot the points found in the table, and connect them using a smooth curve.
Plotting of points forof y = log1/2x Graph ofy = log1/2x
It can be observed that the function is defined only for x > 0. The function is strictly decreasing, and attains all real values. As x approaches 0 from the right, the function increases without bound, i.e., the line x = 0 is a vertical asymptote.
In general, the graphs of y = log
b
x, where b > 0 and b 1 are shown below.
y = logbx (b > 1) y = logbx (0 < b < 1)
1 2 3 4 5 6 7 8 x
−3
−2
−1 1 2 3 4 5 y
O 1 2 3 4 5 6 7 8 x
−3
−2
−1 1 2 3 4 5 y
O
x y
O
D E P E D C O P Y
PROPERTIES OF LOGARITHMIC FUNCTIONS:
PROPERTIES OF LOGARITHMIC FUNCTIONS:
1. The domain is the set of all positive numbers, or {x
| x > 0}.2. The range is the set of all positive real numbers.
3. It is a one-to-one function. It satisfies the Horizontal Line Test.
4. The x-intercept is 1. There is no y-intercept.
5. The vertical asymptote is the line x = 0 (or the y-axis). There is no horizontal asymptote.
Relationship Between the Graphs of Logarithmic and
Relationship Between the Graphs of Logarithmic and Exponential FunctionsExponential Functions Since logarithmic and exponential functions are inverses of each other, their graphs are reflections of each other about the line y = x, as shown below.
y = bx and y = logbx (b > 1) y = bx and y = logbx (0 < b < 1)
Example 3.
Example 3. Sketch the graphs of y = 2log2x. Determine the domain, range, vertical asymptote, x-intercept, and zero.
Solution.
The graph of y = 2log2x can be obtained from the graph of y = log2x by multiplying each y-coordinate by 2, as the following table of signs shows.
x 1/16 1/8 ¼ ½ 1 2 4 8
log2x –4 –3 –2 –1 0 1 2 3
y =2log2x –8 –6 –4 –2 0 2 4 6
D E P E D C O P Y
The graph is shown below.
Analysis:
a. Domain: {x | x
, x > 0}b. Range : {y | y
}c. Vertical Asymptote: x = 0 d. x-intercept: 1
e. Zero: 1
Example 4.
Example 4. Sketch the graph of y = log3x – 1.
Solution.
Sketch the graph of the basic function y = log3x. Note that the base 3 > 1.
The “–1” means vertical shift downwards by 1 unit.
Some points on the graph of y = log3x are (1,0), (3,1), and (9,2).
Shift these points 1 unit down to obtain (1, –1), (3,0), and (9,1). Plot these points.Plot these points.
D E P E D C O P Y
The graph is shown below.
Analysis:
a. Domain: {x | x
, x > 0}b. Range: {y | y
}c. Vertical Asymptote: x = 0 d. x-intercept: 3
The intercept can be obtained graphically. Likewise, we can solve for the x-intercept algebraically by setting y = 0:
0 = log3x – 1 log3x = 1 x = 31 = 3 e. Zero: 3 Example 5.
Example 5. Sketch the graph of y = log0.25(x + 2).
Solution.
Sketch the graph of the basic function y = log0.25x.Note that the base 0 < 0.25 < 1.
Rewrite the equation, obtaining y = log0.25[x – ( –2)].
The “–2” means a horizontal shift of 2 units to the left.
Some points on the graph of y = log0.25x are (1,0), (4, –1), and (0.25,1).
D E P E D C O P Y
Shift these points 2 units to the left to obtain ( –1,0), (2, –1), and ( –1.75,1). Plot thesePlot these points.
points.
Graph:
Analysis:
a. Domain: {x | x
, x > –2}(The expression x+2 should be greater than 0 for log0.25(x+2) to be defined. Hence, x must be greater than –2.)
b. Range : {y | y
}c. Vertical Asymptote: x = –2 d. x-intercept: –1
e. Zero: –1
The examples above can be generalized to form the following guidelines for graphing transformations of logarithmic functions:
Graph of f(x) = a
Graph of f(x) = a loglogbb(x –(x – c) + d c) + d
The value of b (either b > 1 or 0 < b < 1) determines whether the graph is increasing or decreasing.
The value of a determines the stretch or shrinking of the graph. Further, if a is negative, there is a reflection of the graph about the x-axis.
Based on y = alogbx, the vertical shift is d units up (if d > 0) or d units down (if d < 0), and the horizontal shift is c units to the right (if c > 0) or c units to the left (if c < 0).
D E P E D C O P Y
Solved Examples Solved Examples
Analyze each of the following functions by (a) using the transformations to describe how the graph is related to a logarithmic function y =
, (b) identifying the x-intercept, vertical asymptote, domain and range. (c) Sketch the graph of the function.a.) F(x) =
Solution.
The graph of F(x) is shifted 3 units to the right from the graph of f(x) =
.
Domain: { x
| x > 3 } Range: all real numbers Vertical Asymptote: x = 3 x-intercept: (4, 0)D E P E D C O P Y
b.) G(x) =
Solution.
The graph of G(x) is a vertical shift of 3 units downwards from the graph of g(x) =
.Domain: { x
| x > 0 } Range: all real numbers Vertical Asymptote: x = 0 x-intercept: (0.125, 0)c.) H(x) =
Solution.
The graph of H(x) is a stretch by a factor of 3 from the graph of h(x) =
.Domain: { x
| x > 0 } Range: all real numbersVertical Asymptote: x = 0 x-intercept: (1, 0)
D E P E D C O P Y
The previous examples can be generalized to form the following guidelines for graphing transformations of logarithmic functions:
Lesson 22 Supplementary Exercises Lesson 22 Supplementary Exercises
Analyze each of the following functions by (a) using the transformations to describe how the graph is related to a logarithmic function y =
, (b) identifying the x-intercept, vertical asymptote, domain and range. (c) Sketch the graph of the function.a. F(x) =
b. G(x) =
c. H(x) =
Graph of f(x) = a
Graph of f(x) = a loglogbb(x(x – – c) + d c) + d
The value of b (either b > 1 or 0 < b < 1) determines whether the graph is increasing or decreasing.
The value of a determines the stretch or shrinking of the graph. Further, if a is negative, there is a reflection of the graph about the x-axis.
Based on y = alogbx, the vertical shift is d units up (if d > 0) or d units down (if d < 0), and the horizontal shift is c units to the right (if c > 0) or c units to the left (if c < 0)
D E P E D C O P Y
Topic Test 1 Topic Test 1
1. Find the value of the following logarithmic expressions: [5][5]
a. log4(1/64) b. log1/264
2. Express logx + 2logy – 3loga as a single logarithm. [5][5]
3. Solve for x. [10][10]
a.
b.
4. Solve the inequality
[10][10]5. If a certain sound wave has an intensity of 10-5 W/m2, find its corresponding decibel value.
[10]
[10]
6. Graph the following functions. Label all intercepts and asymptotes. Indicate the
domain and range. [10][10]
a.
b. g(x) = 2
Topic Test 2 Topic Test 2
1. Express lnxy2 as a sum, difference or multiple of logarithms. [5][5]
2. Express log56 as a quotient of logarithms to the base 2. [5][5]
3. Solve for x in the following equations. [10][10]
a.
b.
4. Solve the inequality
[10][10]5. Felix deposited ₱50,000 in an investment that earns 5% interest annually. How many years will his investment be equal to ₱175,000? [10][10]
6. Graph the following functions. Label all intercepts and asymptotes. Indicate the
domain and range. [10][10]
a.