Problem 31 - Technical Mathematics with Calculus

Question

Simplify each expression. $$\log _{10} 10^{x}$$

Step-by-Step Solution

Answer

\(\log_{10}(10^{x}) = x\)

Step 1: Identify the Expression

We need to evaluate or simplify: \(\log _{10} 10^{x}\).

Step 2: Apply Order of Operations

Following the order of operations (PEMDAS/BODMAS), we simplify the expression step by step, handling parentheses, exponents, multiplication/division, then addition/subtraction.

Step 3: Compute the Result

Carrying out the operations systematically, we arrive at the result.

Step 4: State the Result

\(\log_{10}(10^{x}) = x\)

Key Concepts

Mathematical Logarithms

Understanding mathematical logarithms is crucial for simplifying complex expressions involving exponential and logarithmic relationships. A logarithm, often represented as \( \log_{b}(x) \), is essentially the inverse operation to exponentiation. This means that if \( b^y = x \), then \( y = \log_{b}(x)\). The value \(y\) represents the power to which the base \(b\) must be raised to yield \(x\). Logarithms have various properties and rules that make computations more manageable, with one of the fundamental rules being the 'logarithm power rule'.

The logarithm power rule is an incredibly helpful shortcut for simplifying expressions. It states that for any positive number \(a\), and any real number \(p\), \( \log_{a}(a^{p}) = p \). This rule is direct when the base of the logarithm and the base of the power inside the logarithm are the same. Understanding this concept is vital as it is a cornerstone for solving more complicated equations involving logarithms.

Simplifying Expressions

Simplifying expressions is a valuable skill in mathematics, which helps in breaking down complex equations into more manageable forms. The process involves applying mathematical operations and properties to condense an expression into its simplest terms. Simplification can include combining like terms, reducing fractions, or employing exponent and logarithm rules.

When simplifying expressions that contain logarithms, it's important to recognize patterns and apply appropriate logarithmic identities and rules. This not only makes the expressions more straightforward to work with but also prepares students for solving logarithmic equations effectively. Simplifying expressions is not just about making them shorter in length; it's about making them clearer and easier to understand and compute. The ability to simplify correctly can often lead to discovering solutions to problems that might at first seem intractable.

Exponents and Logarithms

The relationship between exponents and logarithms is fundamental in mathematics. Exponents represent repeated multiplication, denoted by \( b^n \), where \(b\) is the base and \(n\) is the exponent. Conversely, logarithms help us find the exponent that a base is raised to reach a number.

The deep connection between exponents and logarithms can be leveraged to solve exponential equations. To elucidate this relationship with an example, \( \log_{10}(10^x) = x \) is based on the principle that an exponent on a base ten can be removed by the application of a base ten logarithm, revealing the power \(x\). This is because the logarithm asks the question of which power the base 10 must be raised to, to achieve the number 10 to the power of \(x\), to which the answer is clearly \(x\). It's essential to understand how these properties interact since they often form the backbone of algebra, calculus, and other fields of mathematical study. By mastering the exponent-logarithm relationship, students can handle more intricate problems involving growth and decay, interest calculations, and logarithmic scales used in scientific measurements.