Here are some properties of exponents, followed by some simple and more difficult examples:

Properties of Logarithms

1. log_aa = 1                                         Example:  log₃3=1
2. log_a1 = 0                                         Example:  log₅1=0
3. log_aa^b=b                                          Example:  log₃3⁵=5
4. a^log_ab=b                                          Example:  3^log₃2=2
5. log_ax^r = (r)log_ax                                Example:  log₅x² = (2)log₅x
6. log_a(xy) = log_a(x)+log_a(y)                   Example:  log_a(5x) = log_a(5)+log_a(x)
7. log_a(x/y) = log_a(x) - log_a(y)                 Example:  log_a(5/x) = log_a(5) - log_a(x)
8. y = log_ax  is the same as  x=a^y Example:  y=log₆x    ->    x=6^y
9. ln(x) = log_ex where e ≈ 2.71828

Examples of logarithms

Simplify the following logarithmic problems

1)  log₂8 = log₂2³ = 3log₂2 = 3(1) = 3

Where properties 1,3, and 5 were utilized

2)  log₄(4x²) = log₄(4)+log₄(x²) = 1+2log₄(x)

Where properties 1, 5, and 6 were utilized

Expand the following logarithm

3) 

Notice that everything in the numerator received a (+) sign in front of the newly expanded logarithm and everything in the denominator received a (-) sign.  Furthermore,

This represents the expanded form of the logarithm.  Keep in mind you may be asked to go the other way, starting with an expanded logarithm and working back to a single logarithm.