Problem 57 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{3}{5} \cdot \log_{7}(100) \).

Step 1: Express the Radical as an Exponent

The expression is \( \log_{7}(\sqrt[5]{100})^{3} \). Start by changing the radical \( \sqrt[5]{100} \) into an exponent. Since the fifth root of a number \( x \) is \( x^{1/5} \), rewrite: \( \log_{7}((100)^{1/5})^{3} \).

Step 2: Apply the Power of a Power Rule

Next, use the rule \( (a^{m})^{n} = a^{m \cdot n} \) to combine the exponents: \( \log_{7}(100^{3/5}) \).

Step 3: Use Logarithm Rules for Exponents

Apply the property of logarithms \( \log_{b}(a^{c}) = c \cdot \log_{b}(a) \) to the expression: \( \frac{3}{5} \cdot \log_{7}(100) \).

Step 4: Final Simplification

Now, write the expression without radicals or exponents as \( \frac{3}{5} \cdot \log_{7}(100) \). This expression is now fully simplified.

Key Concepts

Understanding Radical Expressions

Radical expressions often appear in complex mathematical problems. A radical expression involves roots, such as square roots or cube roots. In this case, the original problem involves the fifth root, noted as \( \sqrt[5]{100} \).
Roots can be converted into exponents for easier manipulation during calculations. The fifth root of a number \(x\) is written using the exponent \(x^{1/5}\). Doing this conversion helps when applying certain mathematical rules and performing simplifications.
When you encounter a radical in an equation, remember: the process is about changing the form, not the value. Understanding this concept will make dealing with radicals much easier, especially in problems requiring simplification.

The Role of Exponents in Mathematics

Exponents play a critical role in mathematics by allowing us to express repeated multiplication in a simpler form. In the context of the problem, converting the fifth root into an exponent helps us apply other mathematical rules, such as the power of a power rule.
Exponents have several key properties:

• \((a^m)^n = a^{m \cdot n}\): This power of a power rule lets you multiply exponents of the same base.
• \(a^{m/n}\): Represents the \(n\)-th root of \(a\) raised to the power of \(m\), which is handy for converting roots to exponents.

This understanding of exponents is crucial for simplifying complex expressions, making them easier to work with and ultimately solve.

Logarithm Rules and Their Application

The beauty of logarithms lies in their ability to transform multiplication into addition, division into subtraction, and powers into products. In this problem, applying logarithm rules helps further simplify the expression with powers involved.
One key rule used in the solution is \( \log_b(a^c) = c \cdot \log_b(a) \). This rule allows us to "bring down" the exponent as a coefficient, thereby converting the expression \( \log_7(100^{3/5}) \) into \( \frac{3}{5} \cdot \log_7(100) \).
This simplification is vital because it removes the complexity of dealing with powers inside the logarithm, making the problem much more straightforward to solve. Understanding and applying logarithm rules are essential tools in tackling a wide array of mathematical challenges.