Problem 53 - Precalculus : Building Concepts and Connections

Question

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log 10^{\sqrt{2}}$$

Step-by-Step Solution

Answer

The expression \(\log(10^{\sqrt{2}})\) simplifies to \(\sqrt{2}\)

Step 1: Identify the base and exponent

This expression is in the form \(\log_b(b^n)\) with \(b = 10, n = \sqrt{2}\)

Step 2: Apply logarithmic rule

According to the logarithmic rule, \(\log_b(b^n)\) simplifies to \(n\). Hence, \(\log(10^{\sqrt{2}})\) simplifies to \(\sqrt{2}\)

Key Concepts

Simplifying Expressions

Simplifying expressions in mathematics allows us to present a complicated expression in its simplest form. This process makes understanding, comparing, and using expressions easier. When simplifying expressions involving logarithms, our goal is to use established mathematical rules to reduce the expression to a basic form.
Here, the expression is \( \log(10^{\sqrt{2}}) \). To simplify:

• We identify the components of the expression.
• We see the expression as \( \log_b(b^n) \), hinting at a logarithmic rule.

Once recognized, we can apply the fundamental theorem of logarithms to express the term in its simplest form, leading to a clean and easy-to-understand result.

Logarithmic Rules

Logarithmic rules are essential tools that help us simplify and solve expressions involving logarithms. When we encounter \( \log(10^{\sqrt{2}}) \), using these rules becomes crucial. A fundamental rule in logarithms is that for any real number \( n \), and base \( b \), \( \log_b(b^n) = n \). This means we can transform any power of a base into a simpler expression if the base of the logarithm matches.
This rule comes from the definition of logarithms, which are essentially just inverse operations to exponentiation. It tells us how many times a base needs to be multiplied by itself to reach a given number. Recognizing this rule allows us to conclude that \( \log(10^{\sqrt{2}}) = \sqrt{2} \), thus simplifying the expression effortlessly.

Exponents

Exponents represent repeated multiplication of a number by itself. They are an integral part of mathematics, especially when dealing with logarithms. The expression \( 10^{\sqrt{2}} \) involves an exponent where 10 is used as the base and \( \sqrt{2} \) is the power or exponent.
Understanding how exponents work is fundamental to comprehending logarithmic expressions. The exponent indicates that the base (10 in this case) is raised to the power of \( \sqrt{2} \). The fundamental property of exponents used in simplifying logarithms is that when a logarithm has the same base as the exponent's base, the exponent can be extracted directly. This relationship shows how exponents and logarithms simplify the complex process of multiplication and division into manageable operations using basic laws.