## Sunday, June 3, 2012

Here's an small portion of the chapter on logarithms from my second book.

### Properties of Logarithms and Logarithmic Equations

Recall that a logarithm is an exponent. Since there are properties for exponents, we would expect there to be properties associated with logarithms. The seven properties of logarithms are as follows:

1. log b 1 = 0 because b0 = 1
2. log b b = 1 because b1 = b
3. log b bx = x because bx = bx
4. b log bx = x because log b x is the exponent that b is raised to get x.
5. log b xy = log b x + log b y
6. log b (x/y) = log b x – log b y
7. log b x y = y log b x

Note that an easy way to remember rules 5 and 6 is to think of the rules for exponents. If you multiply like bases, add the exponents. In rule 5 we have the log of a product, which is the same as the sum of the logs. If you divide like bases, subtract the exponents. In rule 6 we have the log of a quotient, which is the same as the difference of the logs.

Below are some examples using the properties of logarithms.

Examples: Write each as the sum or difference of logarithms.

1. log 3 (3 ∙7)
log 3 (3 ∙7) = log 3 3 + log 3 7 (the log of a product is the sum of the logs, property 5)
log 3 (3 ∙7) = 1 + log 3 7 (property 2)

2.. log (1000xyz)

log (1000xyz) = log 1000 + log x + log y + log z (property 5)
log (1000xyz) = 3 + log x + log y + log z (property 3, log 1000 = log 10 3 = 3)

3. log 3 (x/81)

log 3 (x/81) = log 3 x – log 3 81 (the log of a quotient is the difference of the logs, property 6)
log 3 (x/81) = log 3 x – 4 (log 3 81 = log 3 3 4 = 4)

4. log (2xy/7z)

log (2xy/7z) = log(2xy) – log(7z) (the log of a quotient is the difference of the logs, property 6)
log (2xy/7z) = log 2 + log x + log y – log(7z) (apply property 5 to log(2xy))
log (2xy/7z) = log 2 + log x + log y – log 7 – log z (apply property 6 to log(7z))

Examples: Write each of the following without a radical or an exponent.

1. log (1/3)4

log (1/3)4 = 4log(1/3) (property 7, log b x y = y log b x)

2. log √17
log √17 = log(17) ½ (recall that √x = x ½)
log √17 = (1/2)log(17)

Examples: Write each as the sum and/or difference of logarithms.

1. log x3y4z5

log x3y4z5 = log x3 + log y4 + log z5 (property 5)
log x3y4z5 = 3log x + 4log y + 5log z (the log of a power is the power times the log, property 7)

2. ln (4x2y/z)
ln (4x2y/z) = ln (4x2) + ln y – ln z (properties 5 and 6)
ln (4x2y/z) = ln 4 + ln x2 + ln y – ln z (property 5 to ln (4x2))
ln (4x2y/z) = ln 4 + 2 ln x + ln y – ln z (property 7 to ln x2)

3. log 5√(x3y/z2)
log 5√(x3y/z2) = log (x3/5y1/5/ z2/5)
log 5√(x3y/z2) = log x3/5 + log y1/5 – log z2/5 (properties 5 and 6)
log 5√(x3y/z2) = (3/5)log x + (1/5)log y – (2/5)log z (property 7)