Problem 21 - Algebra and Trigonometry

Question

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } 2^{\log _{2} 37}} & {\text { (b) } 3^{\log _{3} 8}} & {\text { (c) } e^{\ln \sqrt{5}}}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Evaluate Expression (a)

The expression given is \(2^{\log_{2} 37}\). This is an application of the property of logarithms, where \(b^{\log_{b} x} = x\). Using this property, we simplify: \(2^{\log_{2} 37} = 37\).

Step 2: Evaluate Expression (b)

The expression given is \(3^{\log_{3} 8}\). Similarly to Step 1, we use the logarithmic property \(b^{\log_{b} x} = x\). Applying this property simplifies the expression: \(3^{\log_{3} 8} = 8\).

Step 3: Evaluate Expression (c)

The expression involves the natural log: \(e^{\ln \sqrt{5}}\). We apply the property \(e^{\ln x} = x\). Hence, \(e^{\ln \sqrt{5}} = \sqrt{5}\).

Key Concepts

Logarithmic Expressions

Logarithmic expressions involve both an exponent and a logarithm. They help in simplifying expressions where the base of the exponent matches the base of the logarithm. This is clearly seen in expressions like \( b^{\log_{b} x} \). Here, the base \( b \) of the exponent and the logarithm are the same. The property \( b^{\log_{b} x} = x \) is a fundamental rule used to solve such expressions.

This property is immensely useful because it allows us to reduce seemingly complex expressions into simpler forms. For example:

• In the expression \( 2^{\log_{2} 37} \), it simplifies directly to 37.
• Similarly, \( 3^{\log_{3} 8} \) simplifies to 8.

This reduction is due to the cancellation effect that occurs when the logarithm and exponent have identical bases.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is particularly unique as its base is the irrational number \( e \) (approximately 2.71828). When expressions involve \( e \), the mathematical operations can be efficiently simplified using properties of the natural log, such as \( e^{\ln x} = x \).

In the context of the exercise where we evaluate \( e^{\ln \sqrt{5}} \),:\

• The natural log simplifies the expression directly from \( e^{\ln \sqrt{5}} \) to \( \sqrt{5} \).

The elegance of the natural logarithm lies in how it cancels out the exponent when paired with its base \( e \). This makes it extremely useful in calculus and beyond, as many natural processes grow exponentially.

Exponentiation and Logarithms

Exponentiation and logarithms are inverse operations. Understanding this relationship is key to unlocking the power of logarithms. When you see exponentiation, it represents repeated multiplication, often written as \( b^n \), where \( b \) is the base and \( n \) is the exponent.

Logarithms help "undo" exponentiation by answering the question: "To what exponent should the base be raised, to produce a certain number?"

• For instance, \( \log_{b}(b^x) = x \) directly follows from the definition of logarithms.
• Using this, \( 2^{\log_{2} 37} \) simplifies to 37, as seen in the exercise.

This inverse nature of logarithms and exponentials is at the heart of many mathematical solutions, particularly in converting multiplicative processes to additive ones, which is extremely useful for calculations.