In the Derivatives of Exponential and Logarithmic Equations section, you will learn how to take derivatives of exponential and logarithmic functions. Here we will review how to factor expressions that contain common exponential and logarithmic components in their terms. To prepare you for logarithmic differentiation, we will review the properties of logarithms and use them to expand logarithmic expressions.

Factoring Exponential and Logarithmic Expressions

The greatest common factor of an algebraic expression can sometimes be an exponential or logarithmic quantity. When factoring expressions that contain common exponential or logarithmic quantities in each of their terms, these common quantities can be factored from the expression.

In the expression [latex]x^2e^{3x}+7e^{3x}[/latex], since [latex]e^{3x}[/latex] is present in both terms, it can be factored from the expression; the factored result is [latex]e^{3x}(x^2+7)[/latex].

In the expression [latex]3x\ln(5)2^{x}+2\ln(5)2^{x}[/latex], since [latex]\ln(5)[/latex] and [latex]2^{x}[/latex] are present in both terms, they can be factored from the expression; the factored result is [latex]\ln(5)2^{x}(3x+2)[/latex].

Use Properties of Logarithms to Expand Logarithmic Expressions

There are a variety of logarithmic properties that allow us to expand a logarithmic expression.

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. For example:

[latex]\begin{array}{l}{\mathrm{log}}_{b}\left(\frac{6x}{y}\right)\hfill & ={\mathrm{log}}_{b}\left(6x\right)-{\mathrm{log}}_{b}y\hfill \\ \hfill & ={\mathrm{log}}_{b}6+{\mathrm{log}}_{b}x-{\mathrm{log}}_{b}y\hfill \end{array}[/latex]

Remember, however, that we can only expand logarithms with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm.