# Logarithms

Log Rules:

1. logb(mn) = logb(m) + logb(n)
2. logb(m/n) = logb(m) – logb(n)
3. logb(mn) = n · logb(m)

In simpler terms:

• Multiplication in condensed form translates to addition during expansion.
• Division in condensed form translates to subtraction during expansion.
• An exponent can be used as a multiplier at the front.

Expanding and Condensing Log Expressions

There is no standard definition, in this context, for “simplifying”. You have to use your own good sense. If they give you a big complicated thing and ask you to “simplify”, then they almost certainly mean “expand”. If they give you a string of log terms and ask you to “simplify”, then they almost certainly mean “condense”.

Expanding Log Expressions:

• log3(2x) = log3(2) + log3(x)
• log4( 16/x) = log4(16) – log4(x) > log4(16) = 2log4( 16/x ) = 2 – log4(x)
• log5(x3) = 3 · log5(x) = 3log5(x)
• More complex samples here: Expansion Complex Samples

Condensing Log Expressions:

• log2(x) + log2(y) = log2(xy)
• log3(4) – log3(5) = log3(4/5)

• 2 · log3(x) = log3(x2)

• More complex condensed samples here: Condensation Complex Samples

Change of Base

You can evaluate a non-standard-base log by converting it to the fraction of the form “(standard-base log of the argument) divided by (same-standard-base log of the non-standard-base)”

Khan Academy Change of Base Video

Tips and Tricks

Check this out for complicated questions that could be answered easily:  Trick Question Page