Problem 99 - Algebra and Trigonometry

Question

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ 10^{\log \sqrt{x}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression \(10^{\log \sqrt{x}}\) is \(x^{0.5}\)

Step 1: Analyze The Logarithmic Base

Observe that the base of the logarithm isn't explicitly defined, thus it is understood to be 10. This allows us to apply the property of logarithms where the base of the logarithm and base of the exponentiation are identical.

Step 2: Application Of Logarithmic Property

As the base of the logarithm and the base of the exponentiation are the same, that means \(10^{\log_{10} a} = a\). Applying this property to our expression, we get \(\sqrt{x}\)

Step 3: Express \(\sqrt{x}\) using Exponential Notation

The square root of x can also be expressed as an exponentiation, i.e., \(\sqrt{x} = x^{0.5}\)