Evaluate or simplify each expression without using a calculator. 10^{log √x}

Verified step by step guidance

1. Recognize that the expression is \(10^{(\log \sqrt{x})}\), where the base of the logarithm is 10 (common logarithm).
2. Recall the property of logarithms and exponents: \(a^{\log_a b} = b\). Here, the base of the exponent and the logarithm match (both base 10), so this property applies.
3. Rewrite the expression using the property: \(10^{(\log \sqrt{x})} = \sqrt{x}\).
4. Express the square root in exponential form: \(\sqrt{x} = x^{\frac{1}{2}}\).
5. Thus, the simplified form of the expression is \(x^{\frac{1}{2}}\).

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as log(a^b) = b·log(a). Understanding how to manipulate logarithmic expressions, including the use of roots and exponents, is essential for simplifying expressions like log(√x).

Relationship Between Exponents and Logarithms

Exponents and logarithms are inverse operations. For example, 10^(log y) = y when the base of the logarithm matches the base of the exponent. This property allows simplification of expressions like 10^(log √x) directly to √x.

Simplifying Radicals

A radical such as √x can be expressed as an exponent, x^(1/2). Recognizing this helps in rewriting expressions involving roots into exponential form, which can then be combined with logarithmic properties for simplification.