Pre-Calculus Mathematics 12 – 4.1 – Exponents Part 1 Logarithms involve the study of exponents so is it vital to know all the exponent laws. Review of Exponent Laws ( ) ( ) ( ) () Note: ( ) Simplifying Exponential Expressions In order to simplify any exponential expression, we must first identify a common base in the expression and then use our rules for exponents as necessary. Example 1: Simplify x x 3 2 8 4 Example 2: Simplify ( ) ( ) Goal: 1. Simplify and solve exponential expressions and equations
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Pre-Calculus Mathematics 12 – 4.1 – Exponents Part 1
Logarithms involve the study of exponents so is it vital to know all the exponent laws.
Review of Exponent Laws
( ) (
)
(
)
( )
Note: ( )
Simplifying Exponential Expressions
In order to simplify any exponential expression, we must first identify a common base in the expression and
then use our rules for exponents as necessary.
Example 1: Simplify xx 32 84
Example 2: Simplify ( ) ( )
Goal: 1. Simplify and solve exponential expressions and equations
Pre-Calculus Mathematics 12 – 4.1 – Exponents Part 1
Example 3: Simplify
(
)
Example 4: Simplify
Solving Exponential Equations
To solve an exponential equation, use the same principles of simplifying expressions to get a common base
on either side of the equation. If the bases on either side are equivalent, then the exponents must also be
equivalent. This allows us to use
Example 5: Solve
Pre-Calculus Mathematics 12 – 4.1 – Exponents Part 1
Example 6: Solve ( )
( )
( )
Example 7: Solve √
Example 8: Solve | |
Practice: Attached Sheet and Page 159 #1 + 2
4.1 Solving Exponential Equations Practice
1.
2.
3.
4.
5.
6.
7.
8.
9. (
)
10.
Solutions
1. -3 2. 1 3. 10 4. 0 5. ⁄
6. ⁄ 7. ⁄ 8. ⁄ 9. ⁄ 10. ⁄
Pre-Calculus Mathematics 12 – 4.1 – Exponents Part 2
Exponential Functions
An exponential function is any function that has a variable in the exponent and a positive base not equal to
zero.
where b > 0 , b≠1
Exploration: Use your graphing calculator to graph the following functions and determine the basic
properties of any exponential function
,
,
(
)
base 0 < b < 1
All pass through the point:
Horizontal Asymptote:
Domain:
Range:
Example 1: Without technology, graph the following. State the domain, range, intercept(s) and
asymptotes.
Goals: 1. Explore the basic properties of an exponential function 2. Apply exponential functions to solve compound interest problems 3. Apply exponential functions to solve growth and decay problems
Pre-Calculus Mathematics 12 – 4.1 – Exponents Part 2
Example 2: Without technology, graph the following. State the domain, range, intercept(s) and
asymptotes.
(
)
Applications of Exponential Functions
There are a variety of situations where exponential functions can be applied to solve real life problems.
Though these may seem like unique situations, they are all just variations of the basic exponential
Compound Interest: Interest calculated on principle invested and then added to the original investment
(
)
where A =
P =
r =
n =
t =
Example 3: Find the interest earned if $2500 is invested at 8.5%/a compounded semi-annually for 4 years
Practice: Page 160 # 5, 6, 7
Pre-Calculus Mathematics 12 – 4.1 – Exponents Part 2
Example 4: After 10 years, who has more money?
Miley: $5000 invested at 8.5%/a compounded semi-annually
Liam: $7000 invested at 6.25%/a compounded annually
Earthquakes/pH
The Richter scale and a pH scale are just power of 10 scales, meaning in , the base is 10.
Example 5: How many times more powerful is an earthquake measuring 8.4 on the Richter scale compared
to an earthquake measuring 4.2?
Growth and Decay Discrete Growth/Decay –growth/decay occurs at a specified rate over a specific time period.
7.5 growth r = , 15% depreciation r = , population doubles r = population triples r = , radioactive half-life decay r =
( )
F = Final amount I = initial amount r = rate of growth/decay t = total time p = period of growth/decay
Pre-Calculus Mathematics 12 – 4.1 – Exponents Part 2
Example 6: A photocopier is set to reduce an image by 8%. What is the copy size after 4 reductions?
Continuous Growth/Decay – growth/decay occurs in a continuous time period:
Eg: bacteria and plant growth, human population growth, radiation absorption and decay
F = Final amount I = initial amount r = continuous rate of growth/decay per unit time t = total time
Example 7: A bacterial population of 3000 doubles every 40 min. How many bacterial exist after 4 hours.
Practice: Page 162 # 9
Pre-Calculus Mathematics 12 – 4.2 – Logarithms
We have explored the exponential function; now let’s look at the inverse of the exponential function.
Exploration: Graph f(x) and its inverse, then algebraically solve for the inverse of the function.
So if we generalize this for any base
and its inverse is
Key Points:
( , ) ( , )
( , ) ( , )
( , ) ( , )
The inverse of an exponential function is another function called a logarithm.
Exponential form Logarithmic form
Note:
Formal Definition: A logarithm of a number is the exponent (y) to which a fixed value (b) must be
raised in order to get some other number (x)
Goals: 1. Define a logarithmic function, explore its basic properties and transform the function 2. Change a function from logarithmic to exponential form and vice versa and solve for unknowns 3. Find the inverse of a given logarithm or exponential function
Greek:
‘logos’– word/speech/logic
‘arithmes’ – numbers
Pre-Calculus Mathematics 12 – 4.2 – Logarithms
Example 1: Change from exponential to logarithmic form
i) 33 = 27 ii) 104 = 10000
Example 2: Determine the numerical value
i) log464
ii) f(x) = log1/3 27
iii) log6 x = 3 iv) log7 (x+2) = 3
Logarithmic Graphs:
The graph of an exponential function can be used to determine the graph of a logarithmic function and
its basic properties.
When
Key Points:
( , ) ( , )
( , ) ( , )
( , ) ( , )
Practice: Page 167 #1 - 4
Pre-Calculus Mathematics 12 – 4.2 – Logarithms
When
(
)
(
)
Key Points:
( , ) ( , )
( , ) ( , )
( , ) ( , )
Logarithmic Domains
Since the inverse of an exponential function is a logarithmic function, the domain of a logarithmic
function is the range of the corresponding exponential function.
{ | . Remember, the base is:
Example 3: Determine the domain of the following:
i) ii)
Transformations of Logarithms
A logarithm, like any other function, can be transformed using the principles associated with
transforming a function. The transformation will always be in relation to the basic graph,
Example 4: Sketch each function.
i)
Pre-Calculus Mathematics 12 – 4.2 – Logarithms
ii)
Inverse Log Functions
To find the inverse of a log function, always refer to the relationship between logarithms and
exponential functions.
Steps to finding inverse:
1. Reverse
2. Isolate the power or the logarithm
3. Switch: exponential form log form
4. Solve for y
Example 5: Determine the inverse of the following
i) ii)
Practice: Page 168 # 5-10, 13
Pre-Calculus Mathematics 12 – 4.3 – Properties of Logarithms
A logarithm is just the inverse of an exponential function.
Just like there are established and proven rules for exponents, there are established and provable rules
for logarithms.
Rules for Logarithms (The Log Laws) Note:
Product Rule: Example: Simplify
Quotient Rule:
Example: Simplify
Power Rule: Example: Simplify
Goals: 1. Analyze the properties of logarithmic functions to determine the rules for logarithmic functions 2. Use the rules for logarithms to simplify expressions
Pre-Calculus Mathematics 12 – 4.3 – Properties of Logarithms
Change of Base:
Example: Find to 3 decimal places
Example 1: Write
in terms Example 2: Find the exact value of
of log 3 and log 5 √
Example 3: Find the exact value of Example 4: Find the exact value of
Pre-Calculus Mathematics 12 – 4.3 – Properties of Logarithms
Example 5: Expand the following √
Simplifying Logarithmic Functions:
1. Understand rules #1-6 above.
2. Do not make up your own rules for logarithms. Common mistakes:
( )
( )
3. Know how to change from exponential form to logarithmic form, and vice versa.
⇔
4. Look for exponential/power relationships between b and x in .
Example 6: Simplify Example 7: Simplify
Practice: Page 177 # 1 - 3
Pre-Calculus Mathematics 12 – 4.3 – Properties of Logarithms