Problem 64 - Precalculus : Building Concepts and Connections

Question

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log _{b} \sqrt[3]{b}, b > 0, b \neq 1$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(1/3\).

Step 1: Express the cube root as an exponent

We start by expressing \(\sqrt[3]{b}\) as \(b^{1/3}\). So our expression becomes \(\log _{b}b^{1/3}\).

Step 2: Apply the power rule of logarithms

We then use the power rule of logarithms, which states that \(\log_b(m^n) = n \log_b(m)\). Hence, our expression becomes \(1/3 \log _{b}b\).

Step 3: Simplify \(\log _{b} b\)

The value of \(\log _{b} b\) is always 1, because any number raised to the power of 1 gives the number itself. Therefore, our expression simplifies to \(1/3 * 1\).

Step 4: Final Calculation

\(1/3 * 1\) equals \(1/3\). So, that's our final answer.

Key Concepts

Power Rule

The power rule is a fundamental concept in logarithms that simplifies expressions significantly. It states that for any base \( b \), if you are taking the logarithm of a number raised to an exponent \( n \), such as \( \log_b(m^n) \), the exponent can be brought down in front as a multiplier. This transforms the expression into \( n \log_b(m) \).

• Imagine you have \( \log_b(b^2) \). Using the power rule, it becomes \( 2 \log_b(b) \).
• This rule is not limited to integers and can be applied to any real number exponent, like fractions or decimals.
• For example, \( \log_b(b^{1/3}) \) would become \( \frac{1}{3} \log_b(b) \).

By simplifying the exponent part of the logarithm, this rule makes it easier to solve or further reduce expressions.

Exponents

Exponents are used to signify repeated multiplication of a number by itself. They are crucial for expressing and manipulating numbers in compact form. For instance, \( b^{1/3} \) tells us we are dealing with a cube root, as it is equivalent to \( \sqrt[3]{b} \).

• When you see \( b^{n} \), it translates to multiplying \( b \) by itself \( n \) times.
• Fractional exponents like \( b^{1/3} \) indicate a root; here, it stands for the cube root.
• Exponents can also be negative, which represents division, such as \( b^{-1} = \frac{1}{b} \).

Understanding how to work with exponents aids in transforming complex expressions into simpler forms, especially when combined with logarithms.

Simplification

Simplification is the process of reducing expressions to their most basic form without changing their value. In logarithms, simplification often involves applying rules such as the power rule.

• One common method is combining like terms or using known identities, like \( \log_b(b) = 1 \).
• Simplified expressions are easier to interpret and solve, potentially revealing further insights about a problem.
• In our case, \( \log_b(b^{1/3}) \) simplifies using the power rule to \( \frac{1}{3} \log_b(b) \), and knowing \( \log_b(b) = 1 \), it reduces to \( \frac{1}{3} \).

The goal of simplification is clarity, making it easier to work with and understand mathematical expressions.

Expressions

Expressions in mathematics can be simple numbers, or combinations of numbers and symbols depicting operations. They represent a particular value or set of values.

• An expression can include variables, numbers, operators (like addition, multiplication), exponents, and parentheses.
• In our exercise, the expression \( \log_b(\sqrt[3]{b}) \) involved both a logarithm and a radical, which needed simplifying.
• The key with expressions is that they can often be translated into different but equivalent forms, revealing different properties or making further calculations easier.

Working with expressions often involves manipulating them to fit the context of a problem or to simplify calculations.