Exponents and Logarithms

What are the rules for combining logarithms?

There are certain rules for combining logarithms. In the following cases, let a be a positive number that does not equal 0; n be a real number; and u and v be positive real numbers:

Logarithmic Rule 1: loga(uv) = loga(u) + loga(v)

Logarithmic Rule 2: loga(u/v) = loga(u) - loga(v)

Logarithmic Rule 3: loga(u)n = nloga(u)

This can be expressed as follows: In rule one, multiplication inside the log is turned into addition outside the log (and vice versa); in rule two, division inside the log is turned into subtraction outside the log (and vice versa); and in rule three, an exponent on anything inside the log can be moved to the front of the log as a multiplier (and vice versa). But remember, these rules only apply if the bases are the same. For example, because the bases are not the same in loga(u) + logb(v), this expression can’t be simplified.


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