The Three Big Properties of Logarithms

There are three big important properties of logarithms that each come from a property of exponents. Here are the properties with their corresponding exponent properties.
 

Logarithm Properties loga(MN)=logaM+logaN loga(M/N)=logaM-logaN loga(Mp)=plogaM

Exponential Properties axay=ax+y ax/ay=ax-y (ax)p=apx

We can get  each of the logarithm properties from the corresponing exponential property by applying the exponential property to x=log_aM and y=log_aN, taking logs on both sides and using the The Inverse Properties of Logarithms.

Property 1

a^loga^Ma^loga^N=a^loga^M+loga^N

By one of the Inverse Properties of Logarithms the left side is MN. (You can also simply see this by saying to yourself, for example in the first factor, that  you are raising a to the power that it needs to be raised to in order to get M, so what you get must be M. See my article The Inverse Properties of Logarithms for further explanation.) So this gives us

MN=a^loga^M+loga^N.

Then taking logs on both sides it becomes

log_a(MN)=log_a(a^loga^M+loga^N),

and the right side of this is log_aM+log_aN by the other Inverse Properties of Logarithms (You can also  see this by noticing that the right side is asking what power can you raise a to in order to get a to the log_aM+log_aN-th power. See my article The Inverse Properties of Logarithms for further explanation.) So

log_a(MN)=log_aM+log_aN.

Property 2

a^loga^M/a^loga^N=a^loga^M-loga^N

By one of the Inverse Properties of Logarithms the left side is M/N, so this gives us

M/N=a^loga^M-loga^N.

Then taking logs on both sides it becomes

log_a(M/N)=log_a(a^loga^M-loga^N),

and the right side of this is log_aM-log_aN by the other Inverse Properties of Logarithms, so

log_a(M/N)=log_aM-log_aN.

Property 3

(_a^loga^M)^p_=a^ploga^M

By one of the Inverse Properties of Logarithms the left side is _M^p, so this gives us

_M^p_=a^ploga^M_.

Then taking logs on both sides it becomes

log_a(_M^p)₌log_a(_a^ploga^M)_,

and the right side of this is plog_aM by the other Inverse Properties of Logarithms, so

log_a(_M^p)=plog_aM.

A good way to get familiar with how to use these properties is to do two kinds of exercises, one where you need to go from the log of some big complicated expression to an expression involving individual logs, and the other where you do the reverse. Here are some worked out examples of both kinds.

Type 1

For these kinds of problems the instruction is to write the expression in terms of individual logarithms and simplify.

Example 1:

log_a(x³y⁵)

Solution:

log_a(x³y⁵)=log_a(x³)+log_a(y⁵)=3log_ax+5log_ay

Example 2:

Solution:

Example 3:

Solution:

Type 2

For these the instruction is to write the expressionas a single logarithm and then simplify.

Example 1:

Solution:

Example 2:

Solution: