In Exercises 81–100, evaluate or simplify each expression without using a calculator. 10^(log √x)

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions. One key property is that log(a^b) = b * log(a). This means that logarithmic expressions can often be rewritten to make calculations easier, especially when dealing with exponents and roots.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, which can be useful in simplifying expressions. For example, log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This is particularly helpful when evaluating logarithms with bases that are not easily computable.

Exponential and Logarithmic Relationships

Exponential functions and logarithmic functions are inverses of each other. This means that if y = b^x, then x = log_b(y). Understanding this relationship is crucial for simplifying expressions involving both exponentials and logarithms, as it allows us to switch between the two forms effectively.