Problem 84 - Algebra and Trigonometry

Question

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \log 10^{8} $$

Step-by-Step Solution

Answer

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

Step 1: Applying the Power Rule

Recognize that the base of the logarithm is 10, which is the same as the base of the power 10. Therefore, you can apply the power rule of logarithms that states \(\log_b{a^n} = n\log_b{a}\). In this case, \(\log 10^{8} = 8 \log{10}\).

Step 2: Simplifying Further

The expression 8 \(\log{10}\) can be simplified further because the \(\log{10}\) means log base 10 of 10 which is 1 (since \(10^1 = 10\)). Therefore, this simplifies to \(8 \cdot 1 = 8\).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool for simplifying expressions involving exponents within a logarithm. This rule states that if you have a logarithm of a power, specifically \( \log_b{a^n} \), it can be simplified to \( n \cdot \log_b{a} \). This is because the exponent \( n \) can be "pulled down" in front of the logarithm.

Let's look at our example to understand this better. Here we have \( \log 10^8 \). Using the power rule, the expression becomes \( 8 \log 10 \). This transformation helps us break down complex logarithmic expressions, making them much easier to handle.

• This rule is particularly useful when solving mathematical problems involving large exponents.
• It transforms a potential multiplicative problem into a more straightforward additive one.

Logarithmic Base

The base of a logarithm is crucial to understanding and simplifying logarithmic expressions. It refers to the number that is repeatedly multiplied to get a particular result. In the example \( \log 10^8 \), the base is 10, which is often seen in logarithms due to its special simplification properties.

One essential fact about logarithms with base 10, known as common logarithms, is that \( \log_{10} 10 \) equals 1. This is because 10 raised to the power of 1 equals 10. This property simplifies calculations remarkably, as we can substitute \( 1 \) for \( \log_{10} 10 \) in any expression.

• Logarithms with a base 10 are frequent in scientific calculations and everyday contexts.
• Understanding the base helps in predicting how the logarithmic function will work.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra and logarithms. In the context of logarithms, simplifying involves reducing a complex expression into a more manageable or recognizable form. Let's simplify the example expression \( 8 \log 10 \) to aid understanding.

We've established that \( \log_{10} 10 = 1 \). So in the expression \( 8 \log 10 \), we substitute the 1 for \( \log 10 \), yielding \( 8 \times 1 = 8 \). This simple arithmetic transforms the initial expression into a plain numerical value, making it much clearer.

• Simplifying helps in evaluating expressions without a calculator or complex computations.
• It aids in recognizing patterns and relationships between mathematical entities.