Problem 58 - Intermediate Algebra

Question

Write each logarithm without an exponent or a radical symbol. Then simplify, if possible. See Example \(5 .\) $$ \log _{3}(\sqrt{10})^{5} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{5}{2} \cdot \log_{3} (10)\).

Step 1: Recognize the Exponent

The expression given is \( \log_{3} (\sqrt{10})^5 \). Here, \((\sqrt{10})^5\) can be rewritten by recognizing that it is the same as \((10^{1/2})^5\).

Step 2: Simplify the Power

Using the power rule \((x^m)^n = x^{mn}\), we can simplify \((10^{1/2})^5\) to \(10^{5/2}\). This will change the expression to \(\log_{3} (10^{5/2})\).

Step 3: Apply the Power Rule of Logarithms

The power rule of logarithms states \( \log_{b} (a^n) = n \cdot \log_{b} (a) \). By applying this rule, we have \( \log_{3} (10^{5/2}) = \frac{5}{2} \cdot \log_{3} (10) \).

Step 4: Simplify the Expression

Our expression \( \frac{5}{2} \cdot \log_{3} (10) \) is already simplified in terms of not having any exponents or radicals. No further simplification is possible without knowing \(\log_{3} (10)\).

Key Concepts

Exponents

Exponents are a crucial concept in mathematics, often used to express repeated multiplication. When you see an exponent, it signifies how many times you multiply a number, known as the base, by itself. For example, in the expression, \( 3^4 \), the base is 3, and you would multiply 3 by itself 4 times, resulting in \( 3 \times 3 \times 3 \times 3 = 81 \). They simplify expressions and make them easier to handle, especially when dealing with large numbers. Exponents follow specific rules that help condense expressions:- **Product of Powers**: To multiply two exponents with the same base, you add their exponents, i.e., \( a^m \times a^n = a^{m+n} \).- **Power of a Power**: When you raise an exponent to another exponent, you multiply the exponents, i.e., \( (a^m)^n = a^{m\times n} \).- **Power of a Product**: Distribute the exponent over a product within parentheses, i.e., \( (ab)^n = a^n \times b^n \).These rules allow for seamless calculations and simplifications in complex algebraic expressions.

Radicals

Radicals are symbols used to indicate roots, and the most familiar radical is the square root (expressed as \( \sqrt{} \)). However, radicals can represent any root, not just the square root. Radicals often appear in expressions involving roots of numbers and need to be simplified to make calculations easier or to convert them into exponents.To simplify a radical, you convert it to an exponent, using the fact that a square root is equivalent to raising to the power of 1/2. For instance, \( \sqrt{10} \) becomes \( 10^{1/2} \). This conversion makes it easier to apply rules of exponents, particularly the power rule, which allows you to evaluate expressions involving powers and roots efficiently. Another essential point about radicals is rationalization, which is the process of eliminating radicals from the denominator of a fraction. This often involves multiplying the numerator and the denominator by a suitable form of the radical that appears in the denominator.

Power Rule of Logarithms

The power rule of logarithms is a helpful tool that simplifies expressions involving logarithms with exponents. It states that \( \log_{b} (a^n) = n \cdot \log_{b} (a) \). This allows you to move the exponent in the argument of the logarithm to the front, turning it into a multiplication.This rule is derived from the definition of logarithms and properties of exponents. By applying the power rule, problems involving complex logarithms become easier to handle and solve. For example, a logarithmic expression like \( \log_{3}(10^{5/2}) \) simplifies to \( \frac{5}{2} \cdot \log_{3}(10) \). The power rule is a key part of logarithmic manipulation. It is particularly useful when dealing with logarithmic equations or simplifying logarithmic expressions, as it can transform a logarithm with an exponential argument into a more straightforward arithmetic operation.

Simplification in Algebra

Simplification is a fundamental process in algebra that involves reducing complex expressions to their simplest form. This makes mathematical expressions less cluttered and easier to work with. Simplification can involve various techniques and rules, such as combining like terms, reducing fractions, and applying fundamental properties of numbers. In the context of logarithms and exponents, simplification can involve: - Applying the laws of exponents to combine or simplify powers. - Using the power rule of logarithms to move exponents outside the logarithms. - Converting radicals to exponents for more straightforward handling and then applying laws of exponents for reduction. Simplifying expressions not only helps in solving algebraic equations more efficiently but also enhances understanding by portraying the core value or structure of the expression in a much clearer way. Whether dealing with high school algebra or advanced calculus, mastery of simplification techniques is indispensable.