1. Chapter 5: Exponential and Logarithmic Functions 5
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?

2. 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…

3. 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?

4. 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 10 x 10 = = 2 2
log x 10

5. 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
That means, = 29
Though the calculator will give you logs to a bunch of places, round your answers to 4 decimal places

6. 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 = 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

7. 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 → = x

8. 5.4: Common and Natural Logarithmic Functions
Natural logarithms
(or Captain’s Log, star date …)
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.

9. 5.4: Common and Natural Logarithmic Functions
Evaluating Natural Logarithms
Use a calculator to find each value.
ln 0.15
ln 0.15 = , which means e = 0.15
ln 186
ln 186 = , which means e = 186
ln (-5)
Undefined, as it’s not possible for a positive number (e) to somehow yield a negative number.

10. 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 → = x
ex = 5 → ln 5 = x → = x

11. 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)

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

13. 5.4: Common and Natural Logarithmic Functions
Graphs 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
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

14. 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.

15. 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.

16. 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)