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

1 / 16

# Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions - PowerPoint PPT Presentation

Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions. Essential Question: What is the relationship between a logarithm and an exponent?. 5.4: Common and Natural Logarithmic Functions. You’ve ran across a multitude of inverses in mathematics so far...

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions' - questa

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Chapter 5: Exponential and Logarithmic Functions5.4: Common and Natural Logarithmic Functions

Essential Question: What is the relationship between a logarithm and an exponent?

5.4: Common and Natural Logarithmic Functions
• 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…
5.4: Common and Natural Logarithmic Functions
• 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?
5.4: Common and Natural Logarithmic Functions
• 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

x

x

10

=

2

10

=

2

x

log

10

5.4: Common and Natural Logarithmic Functions
• 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 Functions
• Evaluating 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

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

5.4: Common and Natural Logarithmic Functions
• 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

5.4: Common and Natural Logarithmic Functions
• 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.
5.4: Common and Natural Logarithmic Functions
• 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.
5.4: Common and Natural Logarithmic Functions
• 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 Functions
• Assignment
• 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 Functions5.4: Common and Natural Logarithmic FunctionsDay 2

Essential Question: What is the relationship between a logarithm and an exponent?

5.4: Common and Natural Logarithmic Functions
• Graphs of Logarithmic Functions
5.4: Common and Natural Logarithmic Functions
• Transforming 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.
5.4: Common and Natural Logarithmic Functions
• 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 Functions
• Assignment
• 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)