Problem 21 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$\log _{8} 8^{-5.7}$$

Step-by-Step Solution

Answer

The expression simplifies to -5.7.

Step 1: Understand the Logarithmic Identity

First, recall the logarithmic identity that can help simplify the expression. The identity is: \( \log_b b^x = x \). This identity is valid because the base \(b\) raised to the power \(x\) and then taking the logarithm with the same base \(b\) results back in \(x\).

Step 2: Apply the Identity

Apply the identity \( \log_b b^x = x \) to the given expression \( \log_8 8^{-5.7} \). Here, the base \(8\) and the expression inside the logarithm also has base \(8\) raised to a power of \(-5.7\).

Step 3: Simplified Expression

Using the identity from Step 1, we can directly simplify \( \log_8 8^{-5.7} \) to \(-5.7\). The expression inside the logarithm was a power of the base itself, so it simplifies directly to the exponent.

Key Concepts

Understanding Logarithms

Logarithms are a way to express exponents in another form. They answer the question, “To what power must the base be raised, to produce a given number?” In simple terms, if you have an expression like \( \log_b a \), you’re looking for a number \( x \) such that \( b^x = a \).

The key part of logarithms is their role in "undoing" the exponentiation process. Logarithms take an exponent and bring us back to our starting number, known as the original base. This is why one of their primary uses is in solving equations where the variable is an exponent.

A few crucial properties of logarithms to remember are:

• \( \log_b 1 = 0 \) because any number to the power of 0 is 1.
• \( \log_b b = 1 \) since a number raised to the first power is itself.
• \( \log_b(b^x) = x \), which leads us back to the exponent.

These properties make logarithms a powerful tool for simplification and solving complex equations.

Exploring Exponents

Exponents refer to the number of times a number, called the base, is multiplied by itself. For example, in \( 8^3 \), the number 8 is multiplied by itself three times. Exponents are a shorthand for expressing how many times a particular number should be used in a multiplication.

Exponents can be both positive and negative. Positive exponents indicate repeated multiplication, while negative exponents represent repeated division, or equivalently, multiplication by the reciprocal of the base. Thus, \( 8^{-5.7} \) means the reciprocal of 8 is raised to the power of 5.7.

Here's how you can think of various exponent situations:

• Zero Exponents: Any nonzero number raised to the power of 0 is always 1, e.g., \( 8^0 = 1 \).
• Negative Exponents: They suggest an inverse operation. For example, \( x^{-n} = 1/x^n \).
• Fractional Exponents: These indicate root operations, such as \( x^{1/2} = \sqrt{x} \).

Understanding these basic rules about exponents makes working with expressions and equations more manageable.

The Art of Simplification

Simplification in mathematics involves transforming an expression into a simpler or more readable form. The goal is to make calculations easier and to present the expression in the most understandable way.

When applying simplification involving both exponents and logarithms, key identities are essential:

• Logarithmic Identity: \( \log_b b^x = x \). This helps directly simplify expressions like \( \log_8 8^{-5.7} \).
• Exponential Rules: Understand how multiplying and dividing powers, especially with like bases, can reduce complexity, and how inverse operations work with negative exponents.

In the original exercise, \( \log_8 8^{-5.7} \) simplifies directly due to the identity. The logarithm and the base "cancel out," leaving just the exponent \(-5.7\). This process showcases how simplification can convert a complex-looking problem into a straightforward solution.

The act of simplifying is about spotting these identities and relationships so that you can efficiently solve problems. With practice, it becomes easier to recognize these patterns and apply them accurately in different contexts.