### 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**b*^{0 }= 1**2. log**

_{b }*b =*1*because**b*^{1}=*b***3. log**

_{b}*b*^{x}*= x*because*b*^{x}*= b*^{x}**4**

*. b*^{log }_{b}^{x}=*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

log

2.. log (1000

_{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 (1000

*xyz***)**
log
(1000

log (1000

*xyz***) =**log 1000 + log*x*+ log*y*+ log*z*(property 5)log (1000

*xyz***) =**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 (2

*xy*/7*z*)
log
(2

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

*xy*/7*z*) = log 2 + log*x*+ log*y*– log(7*z*) (apply property 5 to log(2*xy*))
log
(2

*xy*/7*z*) = log 2 + log*x*+ log*y*– log 7 – log*z*(apply property 6 to log(7*z*))**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

*x*^{3}*y*^{4}*z*^{5}^{ }log

*x*

^{3}

*y*

^{4}

*z*

^{5 }= log

*x*

^{3}+ log

*y*

^{4}+ log

*z*

^{5}(property 5)

log

*x*^{3}*y*^{4}*z*^{5 }= 3log*x*+ 4log*y*+ 5log*z*(the log of a power is the power times the log, property 7)
2.
ln (4

*x*^{2}*y*/*z*)
ln
(4

*x*^{2}*y*/*z*) = ln (4*x*^{2}) + ln*y*– ln*z*(properties 5 and 6)
ln
(4

*x*^{2}*y*/*z*) = ln 4 + ln*x*^{2}+ ln*y*– ln*z*(property 5 to ln (4*x*^{2}))
ln
(4

*x*^{2}*y*/*z*) = ln 4 + 2 ln*x*+ ln*y –*ln*z*(property 7 to ln*x*^{2})
3.
log

^{5}√(*x*^{3}*y*/*z*^{2})
log

^{5}√(*x*^{3}*y*/*z*^{2}) = log (*x*^{3/5}*y*^{1/5}/^{ }*z*^{2/5})
log

^{5}√(*x*^{3}*y*/*z*^{2}) = log*x*^{3/5 }+ log*y*^{1/5}– log*z*^{2/5 }(properties 5 and 6)
log

^{5}√(*x*^{3}*y*/*z*^{2}) = (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.

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Thanks! I like the math skills too. I tutor kids in math all the time, some like it, most don't.

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