Problem 36 - College Algebra

Question

For the following exercises, use the definition of common and natural logarithms to simplify. $$ \log \left(100^{8}\right) $$

Step-by-Step Solution

Answer

\( \log(100^8) = 16 \).

Step 1: Understand the Problem

The problem is to simplify the expression \( \log(100^8) \) using the properties of logarithms. We need to find a way to express this in simpler terms.

Step 2: Apply Power Rule of Logarithms

The power rule of logarithms states that \( \log_b(a^c) = c \cdot \log_b(a) \). Therefore, we can rewrite \( \log(100^8) \) as \( 8 \cdot \log(100) \).

Step 3: Simplify \( \log(100) \)

Since 100 is equal to \( 10^2 \), we use the logarithm property \( \log(10^2) = 2 \cdot \log(10) \). The common logarithm \( \log(10) \) equals 1, so \( \log(100) = 2 \cdot 1 = 2 \).

Step 4: Complete the Simplification

Substitute \( \log(100) = 2 \) back into the expression \( 8 \cdot \log(100) \) to give \( 8 \cdot 2 = 16 \). Thus, \( \log(100^8) = 16 \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool that helps simplify expressions involving exponents in logarithms. This rule states that for any positive number \(a\), base \(b\), and exponent \(c\), the logarithm \(\log_b(a^c)\) is equal to \(c \cdot \log_b(a)\).
This means you can "move" the exponent out of the logarithm as a multiplier.Let's apply this idea to an example:

• Consider \(\log(100^8)\). Using the power rule, you "take out" the 8, so it becomes \(8 \cdot \log(100)\).

By knowing this rule, we are able to greatly simplify the expression. Instead of dealing with a large exponent, the power rule allows us to work with a much more manageable multiplication problem.

Natural Logarithms

Natural logarithms, denoted by \(\ln\), use the constant \(e\) (approximately 2.718) as their base. This is particularly useful in higher-level mathematics and sciences because \(e\) has unique properties when it comes to calculus and exponential growth.While not directly applied in our example, understanding natural logarithms broadens your mathematical toolkit. Here’s a quick breakdown:

• \(\ln(e) = 1\). This is similar to the fact that \(\log_{10}(10) = 1\).
• Natural logarithms share many properties with common logarithms, such as the power rule and the ability to convert between bases using properties of logarithms.

Grasping natural logarithms strengthens your understanding of exponential functions and natural growth phenomena, which appear frequently in real-world situations.

Common Logarithms

Common logarithms are logarithms with base 10, denoted by \(\log\). These are the most familiar type of logarithms, often found in beginner and intermediate math classes Understanding the properties of common logarithms helps you simplify complex expressions. Let's see the rule in action as we **use** the basic property of common log:

• For instance, since \(10^2 = 100\), we know \(\log(100) = \log(10^2) = 2 \cdot \log(10)\).
• Because \(\log(10)\) equals 1, that calculation further simplifies to \(2 \cdot 1 = 2\).

This simplification ability makes common logarithms very useful in practical situations where base 10 quantities are frequently encountered, such as scientific notation and engineering calculations.