Problem 57 - Intermediate Algebra

Question

Write each logarithm without an exponent or a radical symbol. Then simplify, if possible. See Example \(5 .\) $$ \log _{7}(\sqrt[5]{100})^{3} $$

Step-by-Step Solution

Answer

\( \frac{3}{5} \cdot \log_7(100) \)

Step 1: Apply Power Rule of Logarithms

The power rule of logarithms states that \( \log_b(x^n) = n \cdot \log_b(x) \). Apply this rule to the given logarithm: \( \log_7( (\sqrt[5]{100})^3 ) = 3 \cdot \log_7( \sqrt[5]{100} ) \).

Step 2: Convert Radical to Exponential Form

Next, convert the radical expression \( \sqrt[5]{100} \) to an exponential form. Remember that \( \sqrt[n]{x} = x^{1/n} \). Thus, \( \log_7( \sqrt[5]{100} ) = \log_7( 100^{1/5} ) \).

Step 3: Apply Power Rule Again

Apply the power rule again to the expression \( \log_7( 100^{1/5} ) \). This results in \( \frac{1}{5} \cdot \log_7(100) \).

Step 4: Combine the Results

Combine the results from Steps 1 and 3: \( 3 \cdot \left( \frac{1}{5} \cdot \log_7(100) \right) = \frac{3}{5} \cdot \log_7(100) \).

Key Concepts

Power Rule of Logarithms

Understanding the power rule of logarithms is essential when working with expressions that contain exponents inside a logarithm. The power rule states: \( \log_b(x^n) = n \cdot \log_b(x) \). This rule allows you to take the exponent of the term you are logging and move it in front of the logarithm as a multiplier.
This property is particularly helpful in simplifying expressions because it transforms complex powers into a simpler linear form.
For example, if you have \( \log_7((100)^{3}) \), you can take the 3 outside the logarithm, making it \( 3 \cdot \log_7(100) \).

• First, identify the exponent in the logarithmic expression.
• Apply the power rule to simplify the logarithm.
• Remember that this rule can be applied multiple times for nested exponentiations.

This simplification step helps to make complex logarithmic equations much easier to work with.

Radicals and Exponents

Radicals and exponents are closely linked, and converting between the two is a vital skill in simplifying mathematical expressions.
A radical like \( \sqrt[n]{x} \) can be rewritten in exponential form as \( x^{1/n} \). This conversion is useful because operations with exponents are often more straightforward than those with radicals.
For example, if you have \( \sqrt[5]{100} \), you can express this as \( 100^{1/5} \) in exponential form.
Here are some basic concepts:

• Radicals involve roots, such as square roots \( \sqrt{x} \), cube roots \( \sqrt[3]{x} \), etc.
• Exponents represent repeated multiplication: \( x^2 \) means \( x \cdot x \).
• Convert radicals to exponents to apply rules of exponents, especially when simplifying logarithms.

This conversion is essential for applying the power rule of logarithms effectively and accurately.

Simplification of Logarithms

Simplifying logarithmic expressions can seem daunting, but it involves applying the rules of logarithms precisely. The goal is to make the expression as straightforward as possible, often involving fewer terms or a form that is easier to interpret.
In the given problem, by utilizing the power rule of logarithms and converting radicals into exponential forms, the expression becomes simpler: \( \log_7((\sqrt[5]{100})^3) \) simplifies to \( \frac{3}{5} \cdot \log_7(100) \).
To achieve simplification:

• Use the power rule to handle exponents inside logarithms.
• Convert radicals into exponents for easier manipulation.
• Combine results logically, observing consistent mathematical logic throughout.

The simplification not only reduces the complexity but also prepares the expression for further mathematical operations such as solving equations or evaluating values.