4.5 Logarithmic properties  (Page 3/10)

First we note that the quotient is factored and in lowest terms, so we apply the quotient rule.

Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule, noting that the prime factors of the factor 15 are 3 and 5.

The argument is already written as a power, so we identify the exponent, 5, and the base, $x,$ and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

Expressing the argument as a power, we get ${\log }_3\left(25\right)={\log }_3\left(5^2\right).$

Next we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression $4\ln (x),$ we identify the factor, 4, as the exponent and the argument, $x,$ as the base, and rewrite the product as a logarithm of a power: $4\ln (x)=\ln (x^4).$