Solve each equation. Give solutions in exact form. See Examples 5–9. log_2 (log_2 x) = 1

Verified step by step guidance

1. Start by understanding the equation: \( \log_2 (\log_2 x) = 1 \). This means that the logarithm of \( x \) to the base 2, when taken again as a logarithm to the base 2, equals 1.
2. Recall the property of logarithms: if \( \log_b a = c \), then \( a = b^c \). Apply this to the outer logarithm: \( \log_2 x = 2^1 = 2 \).
3. Now, solve the inner equation: \( \log_2 x = 2 \). Again, use the property of logarithms: if \( \log_b a = c \), then \( a = b^c \).
4. Apply this property to find \( x \): \( x = 2^2 \).
5. Conclude by stating that the solution for \( x \) is the value obtained from the previous step.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The logarithm log_b(a) answers the question: 'To what power must the base b be raised to obtain a?' Understanding how to manipulate and solve logarithmic equations is crucial for solving problems involving logarithms.

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another, which is particularly useful when dealing with logarithms of different bases. The formula states that log_b(a) = log_k(a) / log_k(b) for any positive k. This concept is essential for simplifying and solving logarithmic equations.

Nested Logarithms

Nested logarithms occur when one logarithm is contained within another, such as log_b(log_c(x)). Solving these requires understanding the properties of logarithms and often involves isolating the inner logarithm first. This concept is key to unraveling complex logarithmic equations.