The properties that follow below are derived from the fact that the logarithm is defined as the inverse of the corresponding exponential.

To familiarize ourselves with the properties we will first expand the following logarithms. (Assume all variables are positive.)

**Expand**.

Notice that there is no explicit property that allows us to work with

*n*th roots within the argument of the logarithm. To simplify these, first change the root to a rational exponent then apply the power rule.

**Evaluate**

Expanding is useful for learning the rules and properties associated with logarithms but as it turns out, in practice, condensing down to a single logarithm is the skill that we really need to focus on.

**Rewrite as a single logarithm (condense)**.

**Tip**: When simplifying these down to one logarithm use only one operation at a time and work from left to right. Combining or skipping steps usually leads to mistakes. Do not race, work slowly, and pay attention to notation. Evaluate (Round to the nearest ten thousandths where appropriate).

**Simplify**.

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