Problem 75 - Algebra A Combined Function

Question

Simplify. $$ \log _{8}(8)^{-1} $$

Step-by-Step Solution

Answer

The simplified form is -1.

Step 1: Understand the Problem

You are given the expression \( \log_{8}(8)^{-1} \). Our goal is to simplify this expression using the properties of logarithms.

Step 2: Apply the Power Rule for Logarithms

One of the logarithm rules is the power rule, which states \( \log_{b}(a^n) = n \cdot \log_{b}(a) \). In this expression, we identify \( a = 8 \), \( n = -1 \), and \( b = 8 \). Therefore, we can rewrite the expression as \(-1 \cdot \log_{8}(8) \).

Step 3: Evaluate the Logarithm

We know from the properties of logarithms that \( \log_{b}(b) = 1 \) for any base \( b \). Thus, \( \log_{8}(8) = 1 \). Substitute this into the expression to get \(-1 \times 1 \).

Step 4: Simplify the Expression

Multiply the values to obtain \(-1 \times 1 = -1\). This is the simplified form of the original expression.

Key Concepts

Power Rule of Logarithms

Let's dive into the power rule of logarithms, which is a handy tool for simplifying expressions involving logarithms. The power rule states:

• \( \log_{b}(a^n) = n \cdot \log_{b}(a) \)

In simple terms, when you have a logarithm of a number that is raised to a power, you can "bring down" the exponent as a multiplier. This rule primarily helps in simplifying complex log expressions into easier ones. In our exercise, the expression \( \log_{8}(8)^{-1} \) takes the form of \( \log_{b}(a^n) \). Here, the base \( b \) is 8, the value of \( a \) is also 8, and the exponent \( n \) is -1. Application of the power rule allows us to rewrite this as:

• \( -1 \cdot \log_{8}(8) \)

This simplification is very helpful because it breaks down the original problem into a simpler multiplication task. Use the power rule whenever you see a logarithm of an exponent to make life easier.

Logarithmic Simplification

Logarithmic simplification involves reducing logarithmic expressions to their simplest form. This process often calls for the use of various logarithmic rules and properties. Let's look at a key property used in our exercise: the identity property of logarithms.

• \( \log_{b}(b) = 1 \)

This rule states that the logarithm of a number to its own base is always 1. In our simplified expression from the power rule section, we have \( \log_{8}(8) \). Applying the identity property here, we know that \( \log_{8}(8) = 1 \). Substituting this value into the expression \( -1 \cdot \log_{8}(8) \), we further simplify it to:

• \( -1 \cdot 1 = -1 \)

Ensuring every step follows basic properties makes it easier to handle complex equations. With logarithmic simplification, breaking down components with familiar rules simplifies the comprehension of logarithms.

Exponentiation and Logarithms

Understanding exponentiation and its relation to logarithms is crucial when dealing with simplifications. Exponentiation refers to raising a number to a power, while logarithms are its inverse operation.

• If \( x^a = b \), then by definition, \( \log_{x}(b) = a \)

This inverse relationship means calculating a logarithm essentially asks: "To what power must the base be raised, to obtain a certain number?" For instance, in our exercise, the expression \( (8)^{-1} \) asks, "What is \( 8^{-1} \) in terms of logarithms?" When we apply the power rule and simplify, we indirectly navigate inverse operations between powers and logarithms. Knowing this, helps students understand how the logarithm cancels out the exponent through multiplication. Grasping this concept is vital not only in simplifying logarithmic expressions but also in solving equations in algebra and beyond.