Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log x + 3 log y

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Identify the logarithmic expression given: $\log x + 3 \log y$.

Recall the logarithmic property that allows you to move coefficients as exponents inside the log: $a \log b = \log b^a$.

Apply this property to the term $3 \log y$ to rewrite it as $\log y^3$.

Now the expression becomes $\log x + \log y^3$.

Use the logarithmic property that the sum of logs is the log of the product: $\log a + \log b = \log (a \times b)$, so combine to get $\log (x \times y^3)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow combining or breaking down logarithmic expressions. For example, the power rule states that a coefficient in front of a log can be rewritten as an exponent inside the log: a·log b = log(b^a).

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logs as a single logarithm. This involves applying the product, quotient, and power properties to combine terms into one log with coefficient 1. The goal is to simplify the expression for easier evaluation or further manipulation.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator often requires recognizing common log values or rewriting expressions using known bases and exponents. Simplifying logs using properties can help express them in terms of simpler or known logarithms, enabling exact evaluation rather than decimal approximation.

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