Unit 6 Exponential and Logarithmic Functions (Section A) › uploads › 5 › 7 › 5 › 9 ›...

Pre-Calculus 12Unit 6 – Exponential and Logarithmic

Functions(Section A)

Dr. John Lo

Royal Canadian College

2020-2021

1. Exponential functions• Let’s complete the following table and graph the function:

Unit 6 - Exponential and Logarithmic Functions 2

𝒙 𝒚 = 𝟐𝒙

−3

−2

−1

0

1

2

3

RCC @ 2020/2021

• Now, let’s try again with another function:

• How are these two graphs related?


𝒙𝒚 =

𝟏

𝟐

𝒙

−3

−2

−1

0

1

2

3

RCC @ 2020/2021

• By comparing these graphs (and other related ones), we find that


Function x-intercept y-intercept Equation of asymptote

Domain Range

𝑦 = 2𝑥 None 1 𝑦 = 0 𝑥 ϵ R 𝑦 > 0

𝑦 =1

2

𝑥None 1 𝑦 = 0 𝑥 ϵ R 𝑦 > 0

𝑦 = 4𝑥

𝑦 =5

3

𝑥

𝑦 =1

3

𝑥

𝑦 =2

5

𝑥

RCC @ 2020/2021

• Any function of 𝑥 that can be written in the form 𝑦 = 𝑎𝑥, where 𝑎 > 0, is called an exponential function.

• There are two types of graphs of exponential functions:


𝑦 = 𝑎𝑥

where 𝑎 > 1𝑦 = 𝑎𝑥 where 0 < 𝑎 < 1

RCC @ 2020/2021

• An exponential function 𝑦 = 𝑎𝑥, where 𝑎 > 0 and 𝑎 ≠ 1, has the following basic characteristics:

i. The graph has the 𝑦-intercept at 1.

ii. The graph does not have an 𝑥-intercept.

iii. The graph possesses a horizontal asymptote at 𝑦 = 0.

iv. The domain of the graph is 𝑥 ∈ 𝑅.

v. The range of the graph is 𝑦 > 0.

Unit 6 - Exponential and Logarithmic Functions 6RCC @ 2020/2021

2. Transforming exponential functions• The graph of an exponential function can be transformed in the same ways

as other functions we have seen earlier.

• A transformed exponential function takes the following form:


𝑦 = 𝑐𝑎𝑏 𝑥−ℎ + 𝑘

Vertical stretch

Horizontal stretch Horizontal translation

Vertical translation

RCC @ 2020/2021

• The procedures of obtaining a transformed exponential function from the parent function have been discussed in Unit 2. (Review the notes for details.)

• The mapping of points associated with the transformation is described as:


𝑥, 𝑦 →𝑥

𝑏+ ℎ, 𝑐𝑦 + 𝑘

A point on the parentfunction 𝑦 = 𝑎𝑥

The image point on the transformed

function 𝑦 = 𝑐𝑎𝑏 𝑥−ℎ + 𝑘

RCC @ 2020/2021

• Example: Sketch the graph of 𝑦 = 3 2−𝑥+2 − 4. From the graph, determine the intercepts, the horizontal asymptote, domain and range.


• Example: Given 𝑓 𝑥 = 2𝑥. Determine 𝑔(𝑥).


𝑓(𝑥)𝑔(𝑥)

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• Example: For every meter below the surface of water, the light intensity is reduced by 2.5%. The percent, 𝑃, of light remaining at a depth 𝑑 meters can be modeled by the function: 𝑃 = 100 0.975 𝑑.

a) Graph the function for 0 ≤ 𝑑 ≤ 40.

b) To the nearest percent, how much light remains at a depth of 10 m?

c) To the nearest meter, what is the depth when only 50% of the light remains?


3. Review of solving exponential equations

• An exponential equation contains a power with a variable in the exponent.

• For example

• One strategy to solve exponential equations uses the fact that when two powers with the same base are equal, their exponents are equal.


3𝑥 = 81

3𝑥 = 81 = 34

𝑥 = 4

RCC @ 2020/2021

• It is convenient if you are familiar with the integer exponent table:


Pn P = 1 P = 2 P = 3 P = 4 P = 5 P = 6

n = 0 1 1 1 1 1 1

n = 1 1 2 3 4 5 6

n = 2 1 4 9 16 25 36

n = 3 1 8 27 64 125 216

n = 4 1 16 81 256 625 1296

n = 5 1 32 243 1024 3125 7776

n = 6 1 64 729 4096 15625 46656

n = 7 1 128 2187 16384 78125 279936

n = 8 1 256 6561 65536 390625 1679616

RCC @ 2020/2021

• Example: Solve each equation.

(a) 4𝑥 =1

256

(b) 27𝑥 = 92𝑥−1


• Example: Solve each equation.

(a) 2𝑥 = 832

(b) 1252𝑥+1

=3625


4. Logarithmic functions and laws of logarithms• Graph the function 𝑦 = 2𝑥 on the following grids. Plot also its inverse.


• The term logarithm is used to describe the inverse of a power.

• For example: 𝑦 = 10𝑥 versus 𝑥 = 10𝑦


• By definition, the expression 𝑥 = 10𝑦 can be expressed as

• The value of 10 is called the base of the logarithm.

• In general,

where 𝑏 > 0, 𝑏 ≠ 1, 𝑐 > 0.


𝑦 = log10 𝑥

log𝑏 𝑐 = 𝑎 ↔ 𝑐 = 𝑏𝑎

RCC @ 2020/2021

• Example: Write each exponential expression as a logarithmic expression.

• Example: Write each logarithmic expression as an exponential expression.


33 = 27 5−2 =1

2540 = 1

log7 49 = 2 log41

64= −3 log10

1

10000= −4

RCC @ 2020/2021

• Example: Evaluate each logarithm.

(a) log5 3125

(b) log61

216

(c) log8 232


• There are some useful relations and laws of logarithm:


(Product rule)

(Quotient rule)

(Power rule)

RCC @ 2020/2021

• Example: Write each expression as a single logarithm.

• Example: Write each expression in terms of log 𝑎, log 𝑏, and/or log 𝑐.


log 𝑥 + 3 log 𝑦 log 𝑥 + 2 log 𝑦 − 4 log 𝑧 log2 6 − 3

log𝑎

𝑏2log

𝑎2𝑏1/3

𝑐

RCC @ 2020/2021

• Example: Evaluate each expression.

(a) 3 log9 6 − log9 72

(b) 2 log4 6 − 3 log4 3 + log4 12


• Practice: Complete questions 7, 9, 10, 13, and 15 on the Student Workbook (p.442-445)


5. Analyzing and transforming logarithmic functions• What are the characteristics of a typical logarithmic function?

• Example: Graph 𝑦 = log3 𝑥, and identify its intercepts, asymptotes, domain and range.


• Practice: Graph 𝑦 = log8 𝑥, and state its intercepts, asymptotes, domain and range.


• Transformations can be applied to the graph of a logarithmic function, 𝑦 = log𝑎 𝑥, as follows:

where

• The factor of 𝑐: vertical stretch

• The factor of 𝑏: horizontal stretch

• The factor of ℎ: horizontal translation

• The factor of 𝑘: vertical translation

• The image point is given by:


𝑦 = 𝑐 log𝑎 𝑏 𝑥 − ℎ + 𝑘

𝑥, 𝑦 →𝑥

𝑏+ ℎ, 𝑐𝑦 + 𝑘

RCC @ 2020/2021

• Example: Sketch 𝑦 = log2 𝑥 and 𝑦 = log2(4𝑥 + 8) − 1 on the same grid. How are they related? Identify the intercepts, asymptotes, domain and range of both functions.


• Example: Determine the function 𝑓 𝑥 = log𝑎 𝑥. What is 𝑔(𝑥) if it is related to 𝑓(𝑥) by translations?


𝑓(𝑥)

𝑔(𝑥)

RCC @ 2020/2021

• Practice: Determine the function 𝑓(𝑥).


• Challenging question: Determine the function.


6. Solving logarithmic equations

• A logarithmic equation is an equation that contains the logarithm of a variable.

• For example:

• The laws of logarithms may be used to solve logarithmic equations.


log2 𝑥 + log2 𝑥 = 2

RCC @ 2020/2021

• Example: Solve the equation log3 9𝑥 + log3 𝑥 = 4. Verify the solution.


• Example: Solve log 6𝑥 = log 𝑥 + 6 + log 𝑥 − 1 , then verify each solution.


• Example: Solve 3 = log2 𝑥 − 2 + log2 𝑥, then verify each solution.


• Example: Solve each of the equations below.

(a) 36 = 3 2 𝑥+1

(b) 3𝑥+1 = 6𝑥


• Example: Solve, and verify the following logarithmic equation.

log 2𝑥 − 7 + log 𝑥 − 1 = log 𝑥 + 1 + log 𝑥 − 3


• Practice: Do the questions 9, 10, 11, 12 and 13(b) of SWB (p.468-470)


7. Applications of exponential and logarithmic equations• Exponential and logarithmic functions can be applied in many different

areas such as:

✓Compound interest

✓Exponential growth and decay

✓Future and present values

✓Earthquake and Richter scale

✓Sound intensity and decibel

✓Acidity and pH value


• Example: A principal of $1500 is invested at 4% annual interest, compounded quarterly. To the nearest quarter of a year, when will the amount be $2500?


• Example: The half-life of uranium-235 atoms is approximately 700 million years. An igneous rock from the Jurassic period was sampled to determine its age. The concentration of U-235 atoms in the sample was 81.5% of that found in modern day samples. How old is the rock?


• Example: When light passes through glass, the intensity is reduced by 5%. Determine how many layers of glass are needed for only 25% of light to pass through.


• Example: Determine how many monthly investments of $200 would have to be made into an account that pays 6% annual interest, compounded monthly, for the future value to be $100,000.


• Example: A person borrowed $15,000 to buy a car. The person can afford to pay $300 a month. The loan will be repaid with equal monthly payments at 6% annual interest, compounded monthly. How many monthly payments will the person make?


• Example: An earthquake in Grand Banks Nova Scotia which measured 7.3 on the Richter scale was 240 times as strong as an earthquake near Vancouver Island. Determine the Richter scale strength of the Vancouver Island earthquake.


• Example: The loudness of city traffic is 80 dB and the loudness of a car horn is 110 dB. How many times as intense as the sound of city traffic is the sound of a car horn?


• Example: If tomato juice, with a pH of 4.2, is 75 times as acidic as milk, what is the pH of milk?


8. Natural logarithms

• When we compute the following series:

we found that its value is equal to an irrational number 2.71828182846…

• This value is called the Euler’s number, denoted by 𝑒.


1 +1

1+

1

1 × 2+

1

1 × 2 × 3+

1

1 × 2 × 3 × 4+⋯

RCC @ 2020/2021

• On the other hand, when we investigate compound interests, we find that if the principal is $1 and the interest rate is 100%, the amount after a year and the number of compounding period show the following relationship:


Number of compounding periods per year

Amount ($)

1 (annually) 2

2 (semi-annually) 2.25

12 (monthly) 2.613035…

26 (biweekly) 2.667849…

52 (weekly) 2.692596…

365 (daily) 2.714567…

1000000 2.718280…

RCC @ 2020/2021

• Recall the compound interest formula:

• The data shows that when 𝑛 → ∞,


𝐴 = 1 1 +1

𝑛

𝑛 1

1 +1

𝑛

𝑛

→ 2.7182818… = 𝑒

RCC @ 2020/2021

• The value 𝑒 can be used to define the natural logarithm that is commonly used in science and technology.

• By definition, natural logarithm is a logarithm with base 𝑒:

• Note that natural logarithms satisfy all the laws of logarithms.

1) Product law: ln 𝑥𝑦 = ln 𝑥 + ln 𝑦

2) Quotient law: ln𝑥

𝑦= ln 𝑥 − ln 𝑦

3) Power law: ln 𝑥𝑎 = 𝑎 ln 𝑥


ln 𝑥 = log𝑒 𝑥

RCC @ 2020/2021

• The graphs of 𝑦 = 𝑒𝑥 and 𝑦 = ln 𝑥:


• Example: Use natural logarithms to solve the exponential equation: 25 =5𝑒3𝑥. Write the solution to the nearest thousandth.


• Example: Solve each logarithmic equation. Write the solution to the nearest thousandth, where necessary.

(a) 2 ln 𝑥 + ln 9 = 18

(b) − ln 𝑥 + ln 𝑥 − 2 = ln 𝑥 + 2 − ln 2𝑥


• Example: The population of Japan was estimated to be 126.8 million in 2017. Japan has an average population decline of 0.2% per year. Suppose this trend continues.

a) To the nearest tenth of a million, what will the estimated population be in 2030?

b) To the nearest year, when will the estimated population be less than 120 million?


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