Problem 25 - Precalculus Mathematics for Calculus

Question

Evaluate the expression. (a) \(2^{\log _{2} 37}\) (b) \(3^{\log _{3} 8}\) (c) \(e^{\ln \sqrt{5}}\)

Step-by-Step Solution

Answer

(a) 37, (b) 8, (c) \( \sqrt{5} \)

Step 1: Understand the Exponent Logarithm Rule

The general rule to solve expressions of the type \( a^{\log_{a}(b)} \) is that it simplifies directly to \( b \). This is because the logarithm function \( \log_{a}(b) \) gives the power to which \( a \) must be raised to get \( b \). Therefore, \( a^{\log_{a}(b)} = b \). This rule will be applied to both parts (a) and (b).

Step 2: Evaluate Part (a)

For part (a), the expression is \( 2^{\log_{2}(37)} \). Applying the exponent logarithm rule, \( 2^{\log_{2}(37)} = 37 \).

Step 3: Evaluate Part (b)

For part (b), the expression is \( 3^{\log_{3}(8)} \). Similarly applying the exponent logarithm rule, \( 3^{\log_{3}(8)} = 8 \).

Step 4: Understand the Natural Exponential-Logarithm Identity

The expression \( e^{\ln b} \) simplifies to \( b \) for any positive \( b \). This is because \( \ln b \) is the natural logarithm of \( b \), implying that \( e^{\ln b} = b \). This applies to part (c).

Step 5: Evaluate Part (c)

For part (c), the expression is \( e^{\ln \sqrt{5}} \). By the exponential-logarithm identity, it simplifies directly to \( \sqrt{5} \).

Key Concepts

Exponent Logarithm Rule

The Exponent Logarithm Rule is a fundamental concept in exponential and logarithmic functions, simplifying expressions of the form \( a^{\log_{a}(b)} \). In simple terms, the expression resolves to \( b \). This is because the logarithmic function \( \log_{a}(b) \) determines the power needed for the base \( a \) to produce \( b \). So, raising \( a \) to the power of \( \log_{a}(b) \) precisely concludes with \( b \). For example:

• If you have \( 2^{\log_{2}(37)} \), the base and the logarithm base match, simplifying directly to 37.
• Similarly, \( 3^{\log_{3}(8)} \) simplifies to 8.

Understanding this rule helps solve complex logarithm equations quickly and accurately, making it a powerful tool in mathematics.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a specific logarithm that uses the base \( e \), where \( e \approx 2.71828 \). This irrational number \( e \) is essential in mathematics, often appearing in growth and decay scenarios, such as population growth and radioactive decay. It forms the foundation for natural exponential functions.
When you encounter \( e^{\ln b} \), it directly simplifies to \( b \). This simplification is due to the identity rule in natural logarithms, as \( \ln b \) is the power of \( e \) needed to achieve \( b \). For example:

• If you have \( e^{\ln \sqrt{5}} \), the expression simplifies directly to \( \sqrt{5} \).

Understanding the natural logarithm's behavior is critical for professionals and students working with exponential growth or decay functions.

Properties of Exponents

Properties of exponents are shorthand rules designed to simplify and manage expressions involving powers. These properties are essential in algebra, calculus, and beyond. Here's a brief insight into some key properties:

• Product of Powers Property: \( a^m \times a^n = a^{m+n} \). You add the exponents when multiplying like bases.
• Quotient of Powers Property: \( \frac{a^m}{a^n} = a^{m-n} \) evaluates a division by subtracting exponents.
• Power of a Power Property: \( (a^m)^n = a^{m \times n} \) handles nested exponents by multiplying them.
• Zero Exponent Rule: \( a^0 = 1 \) for any \( a eq 0 \). This rule signals that any base to the zero power equals one.

These fundamental properties make working with exponents smoother and more efficient, and their mastery is crucial for advancing in mathematical problem-solving.