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

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

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

- 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…

- 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?

- There are three types of commonly used logs

- 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

- The functions f(x) = 10x and g(x) = log x are inverse functions

x

x

10

=

2

10

=

2

x

log

10

- 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

- 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

- 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.

- (or Captain’s Log, star date 2.71828182846…)

- 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.

- ln 0.15

- Use a calculator to find each value.

- 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

- Assignment
- Page 361, 2 – 36 (even problems)
- Even problems are done exactly like the odd problems, which are in the back of the book)

- Page 361, 2 – 36 (even problems)

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?

- Graphs of 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.

- 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.

- Same as before…

- 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.

- 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.

- Example #2

- 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)

- Page 361, 37 – 48 (all problems)