Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions

Chapter 5: Exponential and Logarithmic Chapter 5: Exponential and Logarithmic FunctionsFunctions5.4: Common and Natural Logarithmic 5.4: Common and Natural Logarithmic FunctionsFunctionsEssential Question: What is the relationship between a logarithm and an exponent?

5.4: Common and Natural Logarithmic 5.4: Common and Natural Logarithmic FunctionsFunctions

You’ve ran across a multitude of inverses in mathematics so far...◦Additive Inverses: 3 & -3◦Multiplicative Inverses: 2 & ½ ◦Inverse of powers: x4 & or x¼

◦But what do you do when the exponent is unknown? For example, how would you solve 3x = 28, other than guess & check?

◦Welcome to logs…

4 x


Logs◦There are three types of commonly

used logs Common logarithms (base 10) Natural logarithms (base e) Binary logarithms (base 2)

◦We’re only going to concentrate on the first two types of logarithms, the 3rd is used primarily in computer science.

◦Want to take a guess as to why I used the words “base” above?


Common logarithms◦The functions f(x) = 10x and g(x) =

log x are inverse functions log v = u if and only if 10u = v

◦All logs can be thought of as a way to solve for an exponent Log base answer = exponent

◦

log

10

10 2x =

x

10 2x =


Common logarithms◦Scientific/graphing calculators have the

logarithmic tables built in, on our TI-86s, the “log” button is below the graph key.

◦To find the log of 29, simply type “log 29”, and you will be returned the answer 1.4624. That means, 101.4624 = 29

◦Though the calculator will give you logs to a bunch of places, round your answers to 4 decimal places

5.4: Common and Natural Logarithmic 5.4: Common and Natural Logarithmic FunctionsFunctionsEvaluating Common Logarithms

◦Without using a calculator, find the following

◦log 1000

◦log 1

◦log

◦log (-3)

If log 1000 = x, then 10x = 1000. Because 103 = 1000, log 1000 = 3

If log 1 = x, then 10x = 1 Because 100 = 1, log 1 = 0

1012 1

2log 10 10 10 10 10 log 10xx

If log (-3) = x, then 10x = (-3) Because there is no real number exponent of 10 to get -3 (or any negative number, for that matter), log(-3) is undefined


Using Equivalent Statements (log)◦Solve each by using equivalent

statements (and calculator, if necessary)

◦log x = 2

◦10x = 29

◦Remember Log base answer = exponent

log x = 2 → 102 = x → 100 = x

10x = 29 → log 29 = x → 1.4624 = x


Natural logarithms◦(or Captain’s Log, star date

2.71828182846…) Common logarithms are used when the

base is 10. Another regular base is used with

exponents, that being the irrational constant e.

◦For natural logarithms, we use “ln” instead of “log”. The ln key is located beneath the log key on your calculator.


Evaluating Natural Logarithms◦Use a calculator to find each value.

ln 0.15 ln 0.15 = -1.8971, which means e-1.8971 = 0.15

ln 186 ln 186 = 5.2257, which means e5.2257 = 186

ln (-5) Undefined, as it’s not possible for a positive

number (e) to somehow yield a negative number.


Using Equivalent Statements (ln)◦Solve each by using equivalent

statements (and calculator, if necessary)

◦ln x = 4

◦ex = 5

◦Remember Log base answer = exponent

ln x = 4 → e4 = x → 54.5982 = x

ex = 5 → ln 5 = x → 1.6094 = x

5.4: Common and Natural Logarithmic 5.4: Common and Natural Logarithmic FunctionsFunctionsAssignment

◦Page 361, 2 – 36 (even problems) Even problems are done exactly like the

odd problems, which are in the back of the book)

Chapter 5: Exponential and Logarithmic Chapter 5: Exponential and Logarithmic FunctionsFunctions5.4: Common and Natural Logarithmic 5.4: Common and Natural Logarithmic FunctionsFunctionsDay 2Day 2Essential Question: What is the relationship between a logarithm and an exponent?

5.4: Common and Natural Logarithmic 5.4: Common and Natural Logarithmic FunctionsFunctionsGraphs of Logarithmic Functions

Exponential functions

Logarithmic functions

Examples

f(x) = 10x; f(x) = ex g(x) = log x; g(x) = ln x

Domain

All real numbers All positive real numbers

Range All positive real numbers

All real numbers

Other f(x) increases as x increases

g(x) increases as x increases

f(x) approaches the x-axis as x decreases

g(x) approaches the y-axis as x approaches 0

'10^x

log x

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

'e^x

ln x

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

5.4: Common and Natural Logarithmic 5.4: Common and Natural Logarithmic FunctionsFunctionsTransforming Logarithmic Functions

◦Same as before… Changes next to the x affect the graph

horizontally and opposite as would be expected Changes away from the x affect the graph

vertically and as expected Example

Describe the transformation from the graph of g(x) = log x to the graph of h(x) = 2 log (x – 3). Give the domain and range. Vertical stretch by a factor of 2 Horizontal shift to the right 3 units Domain: The domain of a log function is all positive

real numbers (x > 0). Shifting three units right means the new domain is x > 3.

Range: The range of a log function is all real numbers. That doesn’t change by transforming the graph.


Transforming Logarithmic Functions◦Example #2

Describe the transformation from the graph of g(x) = ln x to the graph of h(x) = ln (2 – x) - 3. Give the domain and range. x is supposed to come first, so h(x) should be

rewritten as h(x) = ln [-(x – 2)] - 3 Horizontal reflection Horizontal shift to the right 2 units Vertical shift down 3 units Domain: The domain of a log function is all positive

real numbers (x > 0). The horizontal reflection flips the sign, and shifting two units right means the new domain is x < 2.

Range: The range of a log function is all real numbers. That doesn’t change by transforming the graph.

5.4: Common and Natural Logarithmic 5.4: Common and Natural Logarithmic FunctionsFunctionsAssignment

◦Page 361, 37 – 48 (all problems) Problems 37 – 40 only ask to find the

domain, but you may need to figure out the translation first.

Even problems are done exactly like the odd problems, which are in the back of the book)

Documents

Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions