1. More on Logarithmic Functions 9.6

2. Logarithmic Functions
A logarithm is the inverse of an exponential function.
y = bx
x = by
To find the inverse, switch x and y
Exponential function
Logarithmic
function

3. Logarithmic Functions
We write this function like this:
The base of the exponent and log
y = logbx
The answer to a log is a power
The “answer” to by
means
x = by

4. Logarithmic vs Exponential Equations
power
”answer”
y = logbx
base
Rewrite each exponential equation as a log
means
Rewrite each logarithmic equation as an exponential equation
means

5. y = logbx Evaluating Logs
power
Question to ask: What power of b is x?
”answer”
y = logbx
base
Find the value of each log.
Remember, the answer to a log is an exponent!
(because 23 = 8)
(because 3-2 = 1/9)
(because 251/2 = 5)

6. Common and Natural Logs
Log of base 10 (log10x) is called the common log. It is written as log x.
Log in base e (logex) is called the natural log. It is written as ln x. e is a constant like π. Find e on your calculator. What is its value?
The values of common and natural logs can be found with a calculator.

7. Estimating Common and Natural Logs
Estimate using a calculator:

8. More Properties of Logs
Because e 1 = e
Find the exact value:
Because e 0 = 1
The natural log cancels out base e (they are inverses)

9. Solving Logarithmic Equations
power
”answer”
y = logbx
Solve:
base
means
Base is 10
means
Base is e
means
Base is e

10. Change of Base
Some logs have rational values as answers. These are the ones we can do by hand.
(Similar to perfect squares – we don’t need a calculator to find their square root)

11. Change of Base
If the value of the log is irrational, we would need to estimate in order to approximate its value. For example, log29 is irrational because there is no integer value of x such that 2x = 9.
Another problem: the only values of logs we can estimate using a calculator are base 10 and base e (ln).
Is it possible to estimate log29?

12. The change of base formula is:
“Answer” to previous problem a c b c
Old base
New base (any base you want)
Yes! The change of base formula allows us to estimate logs we can’t do by hand.

13. Since we can use any base in the formula, if the objective is to estimate, which bases should we choose?
The natural log
Log in base 10
Use either: OR

14. Verifying solution: 43.3219 is about 100!
Estimate the log using the change of base formula:
3.170
The natural log would also work:
3.170
You try: estimate to 4 decimal places using change of base formula
3.3219
Verifying solution: is about 100!