In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is $1$. $\log 3-3\log x$

Verified step by step guidance

1. Identify the given expression: \(\log 3 - 3 \log x\).
2. Recall the logarithmic property that allows you to move coefficients as exponents: \(a \log b = \log b^{a}\).
3. Apply this property to the term \(-3 \log x\), rewriting it as \(\log x^{-3}\).
4. Rewrite the expression using the property: \(\log 3 - \log x^{3}\).
5. Use the logarithmic subtraction property: \(\log a - \log b = \log \left( \frac{a}{b} \right)\) to combine into a single logarithm: \(\log \left( \frac{3}{x^{3}} \right)\).

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to combine or break 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.

Power Rule of Logarithms

The power rule states that a coefficient multiplied by a logarithm can be expressed as the logarithm of the argument raised to that coefficient. For instance, a·log_b(x) = log_b(x^a). This is essential for rewriting expressions to have a single logarithm.

Combining Logarithmic Expressions

To condense multiple logarithmic terms into one, use the product rule (log_b(M) + log_b(N) = log_b(M·N)) and quotient rule (log_b(M) - log_b(N) = log_b(M/N)). Applying these rules helps write the expression as a single logarithm with coefficient 1.