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
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 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)
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)
I like the math skills which you have mentioned.I can't understand why kids hate math.It's such a interesting subject.If they have any problem in it then they can understand it with help of online tutoring.
ReplyDeleteclass 9cbse sample papers
Thanks! I like the math skills too. I tutor kids in math all the time, some like it, most don't.
ReplyDelete