Properties of Logarithms

Properties of Logarithms

Properties are based off of the rules of exponents (since exponents = logs)

The base of the logarithm can not be equal to 1 and the values must all be positive (no negatives in logs)

𝑏 = 1𝑏 > 0

Product Rule

logbMN = LogbM + logbN

Ex: logbxy = logbx + logby

Ex: log6 = log 2 + log 3

Ex: log39b = log39 + log3b

Quotient Rule

Ex:

Ex:

Ex:

yxy

x555 logloglog

P

MN2log

NMN

Mbbb logloglog

5loglog5

log 222 aa

PNM 222 logloglog

Power Rule

Ex:

Ex:

Ex:

Ex:

BB 5

2

5 log2log

43

7log ba

MxM b

x

b loglog

5log5log 22 xx

ba 77 log4log3

𝑙𝑛 𝑥 =1

2ln(𝑥)

Pre-Req to Solving Equations: Simplifying, Expanding, and Condensing. Let’s try condensing first…

16log4log 44 nm 22 log4log2 2log5log

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

16log4log 44 2log5log nm 22 log4log2

Let’s try something more complicated . . .

Condense the logs

log 5 + log x – log 3 + 4log 5

)xlogx(logxloglog 53525 4444

Let’s try something more complicated . . .

Condense the logs

log 5 + log x – log 3 + 4log 5

= log5𝑥

3+ 𝑙𝑜𝑔54

=log(5𝑥54

3) = log(

𝑥55

3)

𝑙𝑜𝑔4(5

𝑥2) + 𝑙𝑜𝑔4 3𝑥 5 − 𝑙𝑜𝑔4 5𝑥 5

𝑙𝑜𝑔4 5 3𝑥 5 /(𝑥2 5𝑥 5)

)xlogx(logxloglog 53525 4444

Using Properties to Expand Logarithmic Expressions

Expand:

Use exponential

notation

Use the product rule

Use the power rule

2

1

2 2

1

2 2

log

log

log log

12log log

2

b

b

b b

b b

x y

x y

x y

x y

Now we’ll try expanding…

Expand

2

4

y3

x10log

3

85

x2log

Expand

2

4

y3

x10log

3

85

x2log

More Properties of Logarithms

NMNM aa loglog then , If

NMNM aa then ,loglog If

This one says if you have an equation, you can take

the log of both sides and the equality still holds.

This one says if you have an equation and each side

has a log of the same base, you know the "stuff" you

are taking the logs of are equal.