Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$\log 10^{8}$$

Step-by-Step Solution

Answer

The value of the expression \(\log 10^{8}\) is 8.

Step 1: Recognizing the Base

Here, we are dealing with a base 10 logarithm, also known as the common logarithm. The number inside the log is \(10^{8}\) that is 10 raised to the power 8.

Step 2: Applying the Logarithm Rule

A rule of logarithm states that for any positive integer \(a\) and \(n\) : \(\log_{a} (a^{n}) = n\). Application of this rule will give us: \(\log_{10} (10^{8}) = 8\)

Step 3: Final Answer

So, \(\log 10^{8} = 8\).

Key Concepts

Logarithmic Rules

Logarithmic rules are essential tools in mathematics, helping simplify complex expressions. One powerful rule is the basic property of logarithms:

• For any positive integer \(a\) and exponent \(n\), the rule is: \(\log_{a} (a^{n}) = n\).

This means when you have a logarithm with a base that matches the base of the exponent, the exponent is the answer. For example, \(\log_{10} (10^{8})\) simplifies directly to 8. This rule is particularly useful in quickly reducing expressions without needing a calculator.Remember, logarithms are the 'inverse' of exponents; they tell you the power to which a number (the base) is raised to get another number.

Exponential Expressions

Exponential expressions involve numbers raised to a power. Understanding these expressions is vital to grasping other mathematical concepts, including logarithms. An exponential expression is structured as \(a^n\), where:

• \(a\) is the base.
• \(n\) is the exponent or power.

In the expression \(10^8\), 10 is the base and 8 is the exponent, meaning 10 is multiplied by itself 8 times. Recognizing the structure of exponential expressions helps in applying logarithmic rules effectively.These expressions are commonly used in scientific notation, data analysis, and real-world measurements, making them critical for both academic and practical applications.

Base 10

The common logarithm uses base 10, often simply written as \(\log\) without the base specified. Base 10 is familiar because it mirrors our decimal system. Whenever you see \(\log 10^8\), it implies a logarithm with base 10, even if not explicitly shown.Base 10 logarithms are user-friendly:

• Easily relatable due to the decimal system.
• Simplified calculations for powers of 10.

In the case of \(\log 10^8\), the calculation is straightforward because the base of the logarithm matches the base of the exponential expression, leading directly to the answer. Base 10 logarithms serve as a bridge between different scales and are extensively used in fields like science and engineering.