Problem 33 - Algebra and Trigonometry

Question

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{64} 8 $$

Step-by-Step Solution

Answer

The value for \(\log_{64} 8\) is \(0.5\).

Step 1: Rewrite in Exponential Form

Let's denote \(\log _{64} 8 = a\), then by the definition of the logarithm, the equation can be re-written as \(64^a = 8\).

Step 2: Simplify The Expression

Since \(64\) and \(8\) are powers of \(2\), they can be rewritten as \(2^6^a = 2^3\). To simplify these equalities, the bases (\(2\)) need to be the same, the exponents must also be equal. So we should equate \(6a\) and \(3\).

Step 3: Solve for a

We want to find the value for a. So \(6a = 3\), which means \(a = \frac{3}{6} = 0.5\).

Key Concepts

Logarithm Properties

Logarithms are a vital part of mathematics, often used to solve equations involving exponential relationships. One of the pivotal aspects of logarithms is their properties, which allow us to manipulate and simplify logarithmic expressions effectively.

One fundamental property is the power rule, which states \( \log_b(m^n) = n \log_b(m) \). This property is handy when evaluating expressions like \( \log_{64} 8 \) because we often encounter numbers that are powers of the same base. In the exercise, recognizing that 64 and 8 can both be expressed as powers of 2 is the key to simplifying the expression. Another essential property is the change of base, which wasn't directly used in our exercise but is worth mentioning. It allows you to convert a logarithm to a different base, usually to simplify calculations or when a particular base is required for a formula. Understanding these properties is crucial for solving logarithmic equations and for a deeper understanding of how logarithms work in diverse mathematical applications.

Exponential Form

Transforming a logarithmic expression into its exponential form is an essential skill when working with logs. The exponential form provides a different perspective on the relationship encapsulated by a logarithm and is often more intuitive to solve.

Every logarithmic equation \( \log_b(x) = y \) can be equivalently written as \( b^y = x \), where \(b\) is the base, \(y\) is the exponent, and \(x\) is the result. In our example, \( \log _{64} 8 = a \) is rewritten as \( 64^a = 8 \). By translating the logarithmic equation into its exponential counterparts, solving for \(a\) becomes a matter of recognizing the common base and equating exponents, which is a technique that leverages our natural understanding of how exponents work. This step is central to simplifying the original expression and exposes the underlying exponential relationship between the numbers involved.

Simplifying Expressions

The art of simplifying expressions in mathematics is akin to removing the outer layers of an onion to reveal the core. It is about breaking down the complex parts of a mathematical statement into smaller, more manageable pieces.

When simplifying expressions involving exponents and logarithms, recognizing and utilizing the common bases significantly streamlines the process. For example, in our given expression \( \log _{64} 8 \) recognizing that \(64 = 2^6\) and \(8 = 2^3\) transforms a seemingly daunting task into a simple equation - in this case, \(2^{6a} = 2^3\). This revelation strips away the complexity, leaving a direct route to the solution: solving for \(a\) by setting \(6a = 3\), yielding \(a = 0.5\). This simplification ultimately paves the way for clearer understanding and is a testament to the elegance and practicality of mathematics.