1. Review of Logarithms

2. Review of Logarithms
On Exam I: By some questions I received, I think that some of you have an appalling lack of understanding of some basic rules about how to use logarithms!
Example: Using Ω1(E1) = K(E1)f1, etc.
calculate the ratio: R =
[Ω1(E1 = 3.15)Ω2(E2 = 4.7)]/[Ω1(E1 = 3.14)Ω2(E2 = 4.69)]
or R = (Numerator)/(Denominator)
Some of you wrote
ln(R) = ln(Numerator)/ln(Denominator)
which is 100% nonsensical!!
The correct expression is
ln(R) = ln(Numerator) – ln(Denominator)
Also other very basic errors!!

3. Rules of Logarithms If M & N are positive real numbers & b ≠ 1:
Product Rule:
logbMN = logbM + logbN
The log of a product equals the sum of the logs
Examples:
log4(7 • 9) = log47 + log49
log (10x) = log10 + log x
log8(13 • 9) = log8(13) + log8(9)
log7(1000x) = log7(1000) + log7(x)

4. Rules of Logarithms If M & N are positive real numbers & b ≠ 1:
Quotient Rule:
logb(M/N) = logbM - logbN
The log of a quotient equals the difference of the logs
Example:
log[(½)x] = log(x) + log(½)
= log(x) + log(1) – log(2)
But, log (1) = 0, so
log[(½)x] = log(x) - log(2)

5. Rules of Logarithms If M & N are positive real numbers & b ≠ 1
& p is any real number:
Power Rule:
logbMp = p logbM
The log of a number with an exponent equals the product of the exponent & the log of that number
Examples:
log x2 = 2 log x
ln 74 = 4 ln 7
log359 = 9log35

6. log2 (17) = [logb(x)/logb(a)]
Change of Base Formula
Most often we use either base 10 or base e
Most calculators have the ability to do either.
How can we use a calculator to compute the log of a number when the base is neither 10 nor e?
Example:
log2 (17) = ?
Use the formula
So,
log2 (17) = [logb(x)/logb(a)]

7. Basic Properties of Logarithms
Most used properties:

8. Using the Log Function for Solutions
Example
Solve for t:
Take the log of both sides & use properties of logs

9. Properties of the Natural Logarithm
Recall that y = ln x  x = ey
Note that
ln 1 = 0 and ln e = 1
ln (ex) = x (for all x)
e ln x = x (for x > 0)
As with other based logarithms

10. Use Properties for Solving Exponential Equations
Given
Take log of both sides
Use exponent property
Solve for what was the exponent
Note this is not the same as log 1.04 – log 3

11. Common Errors & Misconceptions
log (a+b) is NOT the same as log a + log b
log (a-b) is NOT the same as log a – log b
log (a*b) is NOT the same as (log a)(log b)
log (a/b) is NOT the same as (log a)/(log b)
log (1/a) is NOT the same as 1/(log a)