Taken together, the product rule, quotient rule, and power rule are often called “properties of logs.” Sometimes we apply more than one rule in order to expand an expression.
We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.
With practice, we can look at a logarithmic expression and expand it mentally and then just writing the final answer. Remember, however, that we can only do this with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm.
Rewrite [latex]\mathrm{ln}\left(\frac{{x}^{4}y}{7}\right)[/latex] as a sum or difference of logs with exponent of [latex]1[/latex].
First, because we have a quotient of two expressions, we can use the quotient rule:
We can expand by applying the product and quotient rules.[latex]\begin{array}{lllllll}{\mathrm{log}}_{6}\left(\frac{64{x}^{3}\left(4x+1\right)}{\left(2x - 1\right)}\right)\hfill & ={\mathrm{log}}_{6}64+{\mathrm{log}}_{6}{x}^{3}+{\mathrm{log}}_{6}\left(4x+1\right)-{\mathrm{log}}_{6}\left(2x - 1\right)\hfill & \text{Apply the product and quotient rule}.\hfill \\ \hfill & ={\mathrm{log}}_{6}{2}^{6}+{\mathrm{log}}_{6}{x}^{3}+{\mathrm{log}}_{6}\left(4x+1\right)-{\mathrm{log}}_{6}\left(2x - 1\right)\hfill & {\text{Simplify by writing 64 as 2}}^{6}.\hfill \\ \hfill & =6{\mathrm{log}}_{6}2+3{\mathrm{log}}_{6}x+{\mathrm{log}}_{6}\left(4x+1\right)-{\mathrm{log}}_{6}\left(2x - 1\right)\hfill & \text{Apply the power rule}.\hfill \end{array}[/latex]
Condensing Logarithms
We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined.
How To: Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm
Apply the power property first. Identify terms that are products of factors and a logarithm and rewrite each as the logarithm of a power.
From left to right, apply the product and quotient properties. Rewrite sums of logarithms as the logarithm of a product and differences of logarithms as the logarithm of a quotient.
Use the power rule for logs to rewrite [latex]4\mathrm{ln}\left(x\right)[/latex] as a single logarithm with a leading coefficient of [latex]1[/latex].
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 [latex]4\mathrm{ln}\left(x\right)[/latex], we identify the factor, [latex]4[/latex], as the exponent and the argument, [latex]x[/latex], as the base and rewrite the product as a logarithm of a power: