Solve each equation. Give solutions in exact form. log₂ (log₂ x) = 1

Verified step by step guidance

1. Start by understanding the equation: \(\log_2 (\log_2 x) = 1\). This means the logarithm base 2 of another logarithm base 2 of \(x\) equals 1.
2. Rewrite the equation by setting an inner variable: let \(y = \log_2 x\). Then the equation becomes \(\log_2 y = 1\).
3. Solve for \(y\) by converting the logarithmic equation to its exponential form: \(y = 2^1\).
4. Substitute back \(y = \log_2 x\) to get \(\log_2 x = 2\).
5. Solve for \(x\) by rewriting the logarithmic equation in exponential form: \(x = 2^2\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows you to rewrite logarithmic equations into exponential form to solve for the variable.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving nested logarithmic equations like log_2(log_2 x), you must ensure both the inner and outer logarithm arguments are positive to find valid solutions.

Solving Nested Logarithmic Equations

Solving equations with nested logarithms involves working from the outer logarithm inward. First, isolate the inner logarithm, then solve the resulting simpler logarithmic or exponential equation step-by-step.