lesson 1 logarithmic functions

Lesson 1

Logarithmic Functions

At the end of this lesson, the learner should be able to

● correctly evaluate logarithms; ● correctly write equations in exponential form to

logarithmic form and vice versa; and● correctly solve word problems involving logarithmic

functions.● correctly construct a table of values and sketch the

graph of a logarithmic function; and● correctly determine the equation of a logarithmic

function given its graph.

Objectives

● How will you evaluate logarithms?

● How will you write equations in exponential form to its

logarithmic form and vice versa?

Essential Questions

Before we formally define a logarithmic function, let us observe the following relationship between exponential and logarithmic equations.

• (Click on the link to see how Exponents and Logarithms Working Together )

https://www.mathsisfun.com/algebra/exponents-logarithms.html

Warm up!

● What happens to the base of an exponential expression after conversion of the equation to logarithmic form?

● What happens to the exponent of an exponential expression after conversion of the equation to logarithmic form?

● How can you convert an exponential equation into logarithmic form?

Guide Questions

Example:The inverse of 𝑓 𝑥 = 5𝑥 is 𝑓−1 𝑥 = log5 𝑥.

1 Logarithmic Functionit is a function which follows the form 𝒇 𝒙 = 𝐥𝐨𝐠𝐛 𝒙, where 𝑥 > 0, 𝑏 > 0. and 𝑏 ≠ 1; it is the inverse of the exponential function

Example:The logarithmic form of 25 = 32 is log2 32 = 5.The exponential form of log3 81 = 4 is 34 = 81.

2 Rewriting Exponential Equations to Logarithmic Equations and Vice Versathe logarithmic form of 𝑥 = 𝑏𝑦 is logb 𝑥 = 𝑦; the exponential form of logb 𝑥 = 𝑦 is𝑥 = 𝑏𝑦.

Example:

The exponential form of log 1 000 = 3 is 103 = 1 000.

3 Common Logarithmlogarithm with a base of 10; written as log 𝑥

Example:

The exponential form of ln 𝑎 = 2 is 𝑒2 = 𝑎.

4 Natural Logarithmlogarithm with a base of 𝒆 (Euler’s number); written as ln 𝑥

Example 1: Convert log28 = 3 into its equivalent exponential form.

Example 1: Convert log28 = 3 into its equivalent exponential form.

Solution:Say that the logarithmic form is log𝑏 𝑥 = 𝑦. It follows that𝑏 = 2, 𝑦 = 3, and 𝑥 = 8. Since its corresponding exponential form is 𝑥 = 𝑏𝑦, let us substitute the values of 𝑦, 𝑏, and 𝑥.

Thus, the equivalent exponential form is 𝟐𝟑 = 𝟖.

Example 2: Evaluate log2 16.


Solution:The expression log216 means that we are looking for the exponent of the base 2 to get the answer 16. Since 24 = 16, it follows that log216 = 𝟒.


Solution:Alternatively, we may solve the problem this way. Let 𝑥 be the value of log2 16. It follows that 𝐥𝐨𝐠𝟐 𝟏𝟔 = 𝒙. We can solve for the value of 𝑥 using its exponential form.

log2 16 = 𝑥

2𝑥 = 162𝑥 = 24


Solution:Since 2𝑥 = 24, it follows that 𝑥 = 4 since the bases are equal.

Therefore, log2 16 = 𝟒.

Example:The table of values and the graph of the logarithmic function 𝑓 𝑥 = log5 𝑥 is shown on the next slide.

5 Logarithmic Functionit is a function which follows the form 𝒇 𝒙 = 𝐥𝐨𝐠𝒃 𝒙, where 𝑥 > 0, 𝑏 > 0, and 𝑏 ≠ 1; it can be described using a table of values, an equation, or a graph

Example:


x1

625

1

125

1

25

1

51 5 25 125 625

y −4 −3 −2 −1 0 1 2 3 4

Example 1: Sketch the graph of the function 𝑓(𝑥) = log2 𝑥.


Solution:First, let us construct a table of values. Let 𝑦 = 𝑓(𝑥). We can determine the table of values for the logarithmic function by rewriting the equation 𝑦 = log2 𝑥 into its equivalent exponential form, which is 𝟐𝒚 = 𝒙.

𝒙1

16

1

8

1

4

1

21 2 4 8 16

𝒚 −4 −3 −2 −1 0 1 2 3 4


Solution:Plot the points on the Cartesian plane and connect them using a smooth curve.

The graph of 𝑓 𝑥 = log2 𝑥 is as follows.

Example 2: Sketch the graph of the functions 𝑓(𝑥) = log3 𝑥and 𝑔(𝑥) = 3𝑥 on the same Cartesian plane. Then, compare their graphs, domains, and ranges.


Solution: First, let us construct the tables of values.Let 𝑦 = 𝑓(𝑥). The table of values for 𝑓 𝑥 = log3 𝑥 can be determined by rewriting 𝑦 = log3 𝑥 to its corresponding exponential equation 𝟑𝒚 = 𝒙.

𝒙1

27

1

9

1

31 3 9 27

𝒇(𝒙) −3 −2 −1 0 1 2 3


Solution: The table of values for 𝑔 𝑥 = 3𝑥 is as follows.

𝒙 −3 −2 −1 0 1 2 3

𝒈(𝒙)1

27

1

9

1

31 3 9 27


Solution: Plot the points on the same Cartesian plane and then connect them with a smooth curve.


Solution: The graphs of 𝑓(𝑥) are 𝑔(𝑥) are inverses of one another. It is reflected along the line 𝒚 = 𝒙.


Solution: The domain of 𝑓(𝑥) is the set of positive real numbers, and its range is the set of all real numbers. On the other hand, the domain of 𝑔(𝑥) is the set of all real numbers while its range is the set of positive real numbers.


Solution: Thus, it can be said that the domain of a function is the range of its inverse, and the range of a function is the domain of its inverse.

Individual Practice:

1. Convert 43 = 64 into its equivalent logarithmic form and log5 125 = 3 into its equivalent exponential form.

2. Evaluate log3 243.

Individual Practice:

3. Given the function 𝑓 𝑥 = −log4 𝑥, construct a table of values with five ordered pairs and then sketch its graph.

4. Given the function 𝑓(𝑥) = 2log2 𝑥, construct a table of values with five ordered pairs and then sketch its graph.

Group Practice: To be done in team.

5. In the Richter scale, the magnitude 𝑅 of an earthquake is

given by the formula 𝑅 = log𝐼

𝐼0 ,where 𝐼 is the intensity as

recorded by the seismograph and 𝐼0 is the threshold intensity. Convert the given formula to its equivalent exponential form and then get the magnitude of an earthquake whose intensity is 1 000 000 times the threshold intensity.

Group Practice:

5. In the Richter scale, the magnitude 𝑅 of an earthquake is given by the formula 𝑅 = log𝐼

𝐼0 ,where 𝐼 is the

intensity as recorded by the seismograph and 𝐼0 is the threshold intensity. Convert the given formula to its equivalent exponential form and then get the magnitude of an earthquake whose intensity is 1 000 000times the threshold intensity.

Solution: The given function is 𝑅 = log𝐼

𝐼0. This is a common logarithm with the base 10, an exponent R, and

an answer of 𝐼

𝐼0. Thus the equivalent exponential form of this formula is .

Note that the recorded intensity 1 000 000 𝐼0,, substitute the value I and solve for R.

Therefore, the magnitude R of the earthquake is 6.

Group Practice: To be done in team.

6. Under ideal conditions, the number of a particular strain of Escherichia coli (E. coli) bacteria after 𝑡 hours can be modeled by the function 𝐵 𝑡 = 5(22𝑡). Determine the inverse of 𝐵(𝑡)and then sketch its graph.

1 Logarithmic Functionit is a function which follows the form 𝒇 𝒙 = 𝐥𝐨𝐠𝐛 𝒙, where 𝑥 > 0, 𝑏 > 0. and 𝑏 ≠ 1; it is the inverse of the exponential function

2 Rewriting Exponential Equations to Logarithmic Equations and Vice Versathe logarithmic form of 𝑥 = 𝑏𝑦 is logb 𝑥 = 𝑦; the exponential form of logb 𝑥 = 𝑦 is𝑥 = 𝑏𝑦.

3 Common Logarithmlogarithm with a base of 10; written as log 𝑥

4 Natural Logarithmlogarithm with a base of 𝒆 (Euler’s number); written as ln 𝑥


● How do you evaluate logarithms?

● Why are logarithmic functions important?

● What do you think is the relationship between 𝑥 and 𝑦 in the logarithmic equation logb 𝑥 = 𝑦? Is it increasing or decreasing?

● Why is it important to determine the relationship between the logarithmic and exponential functions?

● What are the properties of the graph of a logarithmic function?

lesson 1 logarithmic functions

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