in order to simplify any exponential expression, we must first … · 2014-11-12 · pre-calculus...

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


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


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. ⁄


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


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


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


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


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


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)


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


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


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


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


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: 3 #1


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:



Tougher questions…….




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?


in order to simplify any exponential expression, we must first … · 2014-11-12 · pre-calculus...

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