Solve each equation. $x={\log }_4\sqrt[3]{16}$

Verified step by step guidance

1. Recognize that the equation is given as \(x = \log_{4} \sqrt[3]{16}\), where you need to find the value of \(x\).
2. Rewrite the cube root expression using fractional exponents: \(\sqrt[3]{16} = 16^{\frac{1}{3}}\).
3. Express 16 as a power of 4, since the logarithm base is 4. Note that \$16 = 4^2$, so substitute to get \(16^{\frac{1}{3}} = (4^2)^{\frac{1}{3}}\).
4. Use the power of a power property: \((a^m)^n = a^{m \cdot n}\), to simplify \((4^2)^{\frac{1}{3}}\) to \(4^{\frac{2}{3}}\).
5. Now the equation becomes \(x = \log_{4} 4^{\frac{2}{3}}\). Use the logarithm power rule \(\log_b (a^m) = m \log_b a\) and the fact that \(\log_b b = 1\) to simplify and solve for \(x\).

Key Concepts

Logarithms and Their Properties

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log₄(16) asks, '4 raised to what power equals 16?' Understanding how to interpret and manipulate logarithms is essential for solving equations involving them.

Radicals and Rational Exponents

Radicals like cube roots can be expressed as rational exponents; for instance, the cube root of 16 is 16^(1/3). Converting between radical notation and exponents helps simplify expressions and apply logarithmic rules effectively.

Change of Base and Simplification Techniques

Simplifying logarithmic expressions often requires rewriting numbers as powers of the same base or using properties like log(a^b) = b·log(a). Recognizing how to rewrite 16 and apply these properties is key to solving the given equation.