What are the rulesfor combining logarithms?
There are certain rules for combining logarithms. In the following cases, let abe a
positive number that does not equal 0; nis a real number; and uand vare 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)nnloga(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 any-
thing 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.How do you expand logarithms?
Like an algebraic expression, it is possible to expand logarithms, which is a way of
“picking apart” an expression. The following lists two examples of expanding a loga-
148 rithmic expression:
In this example of an exponential function, when 2 is raised to the power of x, the line representing the function
always lies above the xaxis.