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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
Example 8: Simplify
Example 9: Simplify
Example 10: Simplify
Example 11: Simplify ( )( )
Practice: Page 178 # 4, 5
Pre-Calculus Mathematics 12 – 4.4 – Exponential & Logarithmic Functions
Using the rules we have established for logarithmic functions, we can now start to solve logarithmic
equations. Remember, an equation has an equal sign, so the solution must equal something.
Be careful of logarithmic equations though, the solutions must be a part of the domain of that function
Note:
Steps for solving logarithmic functions:
1. If a constant exists in the equation, bring all the logs to one side, constant(s) on the other.
OR
If no constant exists in the equation, combine logs on each side into single logs with a common
base.
2. Convert to exponential form using
OR
Use rule:
3. Solve the resulting equation for the unknown variable.
4. Reject any extraneous root(s) using: ;
Example 1: Solve for x: ( ) ( )
Goal: 1. Solve logarithmic or exponential equations
Pre-Calculus Mathematics 12 – 4.4 – Exponential & Logarithmic Functions
Example 2: Solve for x: ( ) ( )
Expressing equations in terms of stated or defined variables
Sometimes we have a question that only contains variables and we need to solve the logarithmic
equation in terms of the stated variables. This is just a slight variation on what we have already looked
at, just with no numbers
Example 3: Solve for A in terms of B and C:
Example 3: If , what is in terms of a and b?
Practice: Page 183 #1
Pre-Calculus Mathematics 12 – 4.4 – Exponential & Logarithmic Functions
Remember from the start of this chapter, we were able to solve exponential equations with a common
base.
Now, with logarithms, we can solve an exponential equation that doesn’t have a common base.
Steps for solving Exponential Equations – Variable exponents and no common bases
1. Simplify if possible then log both sides. (base 10)
2. Use log laws to move exponents in front of each log.
3. Expand out the logarithms; bring logs containing unknowns on one side.
4. Isolate and solve for the unknown (usually by factoring), simplify to a single log if possible.
5. Determine a numerical value of the single log (if possible)
Example 4: Solve for x:
Practice: Page 184 # 2 - 4
Pre-Calculus Mathematics 12 – 4.4 – Exponential & Logarithmic Functions
Tougher questions…….
Example 5: Solve for x:
Example 6: Solve for x:
Practice: Page 184 # 5 - 7
Pre-Calculus Mathematics 12 – The Natural Logarithm
Common Logs vs. Natural Logs
Example 1: Solve 2042 x using common logs.
Example 2: Solve 2042 xusing natural logs.
When solving an exponential equation, it does not matter whether it is with common logs or natural
logs. But if the exponential equation has a base of e, always use natural logs.
Example 2: Solve 82 xe
Goal: 1. Solve logarithmic equations with natural logarithms
Practice: See back page
Pre-Calculus Mathematics 12 – The Natural Logarithm
Solve each equation using common logarithms.
1.) 108 x 2.) 204.2 x 3.) 8.198.1 5 x
Solve each equation using natural logarithms,
4.) 8535 x 5.) 2542 x 6.) 426 x
7.) 347 xx
Solve each equation
8.) te 04.01751249 9.)
xe 048.012 10.) 24.8 te
Solutions:
1) x = 1.1073 2) x = 3.4219 3) x = 10.0795 4) x = 0 .8088 5) x = 1.1610 6) x = 2.0860 7) x = 7.4316
8) t = -49.1328 9) x = 51.7689 10) t = 4.1282
Pre-Calculus Mathematics 12 – 4.5 – Applications of Exponential & Logarithmic Functions
Using the rules we have established for exponential and logarithmic functions, we can now start to
apply these to real life situations.
Recall from 4.1:
Compound Interest
(
)
Discrete Growth
( )
Continuous Growth
Example 1: Danny-Boy inherits $10 000 and invests it in a guaranteed investment certificate (GIC) at
6 %. How long will it take to be worth $15 000 it is compounded a) monthly b)
continuously?
Example 2: A major earthquake of magnitude 8.3 is 120 times as intense as a minor earthquake.
Determine the magnitude, to the nearest tenth, of the minor earthquake.
Goal: 1. Apply logarithmic or exponential functions to solve real world problems
Pre-Calculus Mathematics 12 – 4.5 – Applications of Exponential & Logarithmic Functions
Example 3: The half-live of carbon-14 is 5730 years. A bone sample is found to have 49.5% of the C-14
remaining. Determine the age of the bone.
Example 4: The population of a fish in a lake is increasing at a rate of 3.5% per year. How long will it
take the fish population to double?
Practice: Page 192 # 1 - 11