In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ${\log }_4\left(\frac{\sqrt{x}}{64}\right)$

Verified step by step guidance

1. Recall the logarithmic expression given: \(\log_4 \left( \frac{\sqrt{x}}{64} \right)\).
2. Use the logarithm property for division: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\). So rewrite the expression as \(\log_4 (\sqrt{x}) - \log_4 (64)\).
3. Express the square root as an exponent: \(\sqrt{x} = x^{\frac{1}{2}}\). Then apply the power rule for logarithms: \(\log_b (M^p) = p \log_b M\). So \(\log_4 (\sqrt{x}) = \frac{1}{2} \log_4 x\).
4. Rewrite 64 as a power of 4 if possible. Since \$4^3 = 64\(, we have \)\log_4 (64) = \log_4 (4^3)$.
5. Apply the power rule again: \(\log_4 (4^3) = 3 \log_4 4\). Since \(\log_4 4 = 1\), this simplifies to 3. So the expression becomes \(\frac{1}{2} \log_4 x - 3\).

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions by expanding or condensing them. For example, log_b(M/N) = log_b(M) - log_b(N) and log_b(M^p) = p * log_b(M). These properties are essential for simplifying and expanding logarithmic expressions.

Radicals and Exponents

Understanding how to express radicals as fractional exponents is crucial. For instance, the square root of x can be written as x^(1/2). This conversion helps apply logarithmic power rules effectively, enabling the expansion of logarithmic expressions involving roots.

Evaluating Logarithms with Known Bases

Evaluating logarithms without a calculator often involves recognizing numbers as powers of the base. For example, 64 is 4 raised to the 3rd power (4^3). This allows simplification of expressions like log_4(64) by rewriting 64 in terms of the base 4, making evaluation straightforward.