4.5 Logarithmic properties  (Page 2/10)

Let $m={\log }_{b}M$ and $n={\log }_{b}N.$ In exponential form, these equations are $b^m=M$ and $b^n=N.$ It follows that

Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider ${\log }_b(wxyz).$ Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:

Using the quotient rule for logarithms

For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to combine the quotient of exponents by subtracting: $x^{\frac{a}{b}}=x^{a-b}.$ The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.

Given any real number $x$ and positive real numbers $M,$ $N,$ and $b,$ where $b\ne 1,$ we will show

Let $m={\log }_{b}M$ and $n={\log }_{b}N.$ In exponential form, these equations are $b^m=M$ and $b^n=N.$ It follows that

For example, to expand $\log \left(\frac{2x^2+6x}{3x+9}\right),$ we must first express the quotient in lowest terms. Factoring and canceling we get,

Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.