Problem 34 - Algebra for College Students

Question

Evaluate each logarithmic expression. \(10^{\log _{10} 14}\)

Step-by-Step Solution

Answer

The result of the expression is 14.

Step 1: Understand the Expression

The expression we need to evaluate is \(10^{\log_{10} 14}\). The base of the logarithm is 10, and the expression involves the exponential form with base 10 as well.

Step 2: Apply the Logarithm-Exponent Rule

One key property of logarithms is that \(a^{\log_a b} = b\). This property applies here because the base of the logarithm (10) and the base of the exponential function (10) are the same. Thus, we have \(10^{\log_{10} 14} = 14\).

Key Concepts

Logarithm-Exponent Rule

In mathematics, understanding the interplay between logarithms and exponentials is crucial. One powerful tool is the logarithm-exponent rule. This rule states that if you have an expression of the form \(a^{\log_a b}\), it simplifies directly to \(b\). This property signifies that an exponential function with a logarithmic exponent of the same base collapses to the argument of the logarithm. Here's why this works:

• The logarithm \(\log_a b\) asks, "To what power must we raise \(a\) to get \(b\)?"
• When this is seen as an exponent \(a^{\log_a b}\), you're effectively raising \(a\) to the power that brings you back to \(b\).

So, if you see \(10^{\log_{10} 14}\), you can immediately replace it with 14. This simplification can save a lot of time when working with complex expressions.

Exponential Functions

An exponential function is one in which a constant base is raised to a variable exponent. In mathematical notation, it appears as \(a^x\), where \(a\) is the base and \(x\) is the exponent. These functions grow very quickly when \(x\) is positive, and they vanish towards zero when \(x\) is negative. A remarkable property of exponential functions is their natural growth, which makes them occur often in scientific models:

• They describe how living organisms grow under constant conditions.
• Compound interest in finance is another example.
• Radioactive decay can be modeled as an exponential reduction over time.

In essence, the beauty of exponential functions lies in their ability to model both rapidly growing and shrinking processes, making them versatile across various fields of study.

Base of Logarithm

The base of a logarithm is the number that gets raised to a power. It forms an intrinsic part of how logarithms work. Written as \(\log_a b\), \(a\) is the base, which is raised to some power to yield \(b\). Understanding the base is vital because it determines the nature of the logarithm:

• Common logarithms use base 10 and are often seen as \(\log\).
• Natural logarithms use the irrational base \(e\) (approximately 2.718) and are denoted as \(\ln\).
• Binary logarithms use base 2, commonly applied in computer science.

In expressions like \(10^{\log_{10} 14}\), having matching bases between the logarithm and the exponent leads to elegant simplifications, highlighting the base's crucial role in such equations.