8.4 – Properties of 8.4 – Properties of LogarithmsLogarithms

Properties of LogarithmsProperties of Logarithms

There are four basic properties of There are four basic properties of logarithms that we will be working logarithms that we will be working with. For every case, the base of the with. For every case, the base of the logarithm can not be equal to 1 and logarithm can not be equal to 1 and the values must all be positive (no the values must all be positive (no negatives in logs)negatives in logs)

Product RuleProduct Rule

loglogbbMNMN = Log= LogbbM + M + loglogbbNN

Ex: logEx: logbbxy = logxy = logbbx + logx + logbby y

Ex: log6 = log 2 + log 3Ex: log6 = log 2 + log 3

Ex: logEx: log339b =9b = loglog339 + log9 + log33bb

Quotient RuleQuotient Rule

Ex: Ex:

Ex: Ex:

Ex: Ex:

yxy

x555 logloglog

P

MN2log

NMN

Mbbb logloglog

5loglog5

log 222 aa

PNM 222 logloglog

Power RulePower Rule

Ex: Ex:

Ex: Ex:

Ex: Ex:

BB 52

5 log2log

437log ba

MxM bx

b loglog

5log5log 22 xx

ba 77 log4log3

Let’s try someLet’s try some Working backwards now: write the following as a Working backwards now: write the following as a

single logarithm.single logarithm.

16log4log 44 nm 22 log4log2 2log5log

Let’s try someLet’s try some Write the following as a single logarithm.Write the following as a single logarithm.

16log4log 44 2log5log nm 22 log4log2

Let’s try something more Let’s try something more complicated . . .complicated . . .Condense the logsCondense the logs

log 5 + log x – log 3 + 4log 5log 5 + log x – log 3 + 4log 5

)xlogx(logxloglog 53525 4444

Let’s try something more Let’s try something more complicated . . . complicated . . . Condense the logsCondense the logs

log 5 + log x – log 3 + 4log 5log 5 + log x – log 3 + 4log 5

)xlogx(logxloglog 53525 4444

Let’s try something more Let’s try something more complicated . . .complicated . . . ExpandExpand

2

4

y3

x10log

3

8 5

x2log

Let’s try something more Let’s try something more complicated . . . complicated . . . ExpandExpand

2

4

y3

x10log

3

8 5

x2log