Problem 59 - 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. $$10^{\log (3 x+1)}$$

Step-by-Step Solution

Answer

The simplified expression is \((3x+1)\)

Step 1: Recognize the property of logarithms

The expression given: \(10^{\log (3x+1)}\) falls into form of \(a^{\log_a (b)}\), in which the base \(a\) is 10, and \(b\) is \((3x+1)\). According to property of logarithms \(a^{\log_a (b)} = b\), the base of the exponential and the base of the logarithm are the same, so they 'cancel' each other out.

Step 2: Apply the property of logarithms

On applying the property \(10^{\log (3x+1)}\) becomes \((3x+1)\).

Key Concepts

Understanding Logarithmic Properties

When dealing with logarithms, understanding their foundational properties can significantly simplify complex expressions. One such property is the inverse relationship between logarithms and exponentiation. In the context of the given exercise, where we see the expression \(10^{\log (3x+1)}\), this relationship is central to simplifying the problem.

Essentially, a logarithm answers the question: to what power must we raise the base (in this case, 10) to obtain a certain number? The expression \(\log_{10} (3x+1)\) is asking for the power to which 10 must be raised to yield \(3x+1\). When we raise 10 to this exact logarithmic value, the two operations effectively 'cancel out', bringing us back to the original number \(3x+1\). This is an application of the logarithmic property \(a^{\log_a (b)} = b\), which is crucial for simplifying logarithmic expressions.

Exponentiation and its Role in Simplifying Expressions

Exponentiation, the process of raising a number to a power, is deeply connected to logarithms. This connection is at the heart of why certain logarithmic expressions can be simplified so elegantly. In our exercise, \(10^{\log (3x+1)}\) involves exponentiation where 10 is the base and \(\log (3x+1)\) is the exponent.

The key to simplification lies in recognizing that exponentiation and logarithms are inverse operations. Like addition and subtraction, or multiplication and division, exponentiation and logarithms undo each other's effects when applied sequentially. This interrelationship means that an exponential expression with a logarithmic exponent, having the same base as the logarithm, simplifies to just the input of the logarithm. Understanding this allows students to confidently transform expressions without needing a calculator, relying instead on algebraic insight.

Logarithm Rules: A Toolset for Simplification

Logarithm rules are the algebraic toolset students use to manipulate and simplify logarithmic expressions. These rules include the product rule, quotient rule, power rule, and the changing of base formula, among others. However, for the exercise in question, the most relevant rule is what's often called the 'exponentiation-logarithm identity'.

This identity is precisely what was used to simplify \(10^{\log (3x+1)}\) to \(3x+1\). It states that a number raised to the power of its logarithm (with respect to the same base) is simply the argument of the logarithm, as the operations counteract each other. By mastering this and other logarithm rules, students are equipped to handle a variety of problems, from the simple to the complex, and to simplify expressions that might at first glance seem daunting.