Problem 29 - Algebra A Combined Function

Question

Find the exact value of each logarithm. $$ \ln \sqrt{e} $$

Step-by-Step Solution

Answer

The exact value of \( \ln \sqrt{e} \) is \( \frac{1}{2} \).

Step 1: Understand the Problem

The problem asks us to find the natural logarithm of the square root of the mathematical constant \( e \), where \( e \) is approximately 2.71828. This is expressed as \( \ln\sqrt{e} \).

Step 2: Simplify the Expression

Recognize that the square root can be expressed as an exponent. The square root of \( e \) can be rewritten as \( e^{1/2} \). So, \( \ln\sqrt{e} = \ln e^{1/2} \).

Step 3: Apply Logarithmic Property

Utilize the property of logarithms that states \( \ln a^b = b \cdot \ln a \), where \( a \) is the base and \( b \) is the exponent. Apply this rule to obtain \( \ln e^{1/2} = \frac{1}{2} \cdot \ln e \).

Step 4: Evaluate \( \ln e \)

Recognize that \( \ln e = 1 \) because the natural logarithm of \( e \) is always 1. This gives us \( \frac{1}{2} \cdot 1 \).

Step 5: Calculate the Result

Multiply \( \frac{1}{2} \) by 1 to obtain \( \frac{1}{2} \).

Key Concepts

Logarithmic Properties

Logarithmic properties are the rules that govern how we manipulate and simplify logarithmic expressions. For instance, when you encounter an expression like \( \ln a^b \), you can apply the power rule for logarithms. This property allows us to bring down the exponent \( b \) in front of the logarithm as a multiplier, resulting in \( b \cdot \ln a \).
This makes it easier to work out complex logarithmic expressions without a calculator. Another important property is that multiplying numbers inside a logarithm equates to adding logarithms: \( \ln(ab) = \ln a + \ln b \). Similarly, dividing numbers inside a logarithm means subtracting one logarithm from the other: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \).
By mastering these properties, you can simplify logarithmic problems more efficiently, saving you time and reducing potential calculation mistakes.

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. When you see a number raised to a power, such as \( e^{1/2} \), it's expressing that the base \( e \) is multiplied by itself one half times, which effectively means taking the square root of \( e \).
Exponents are used in various mathematical applications, providing a succinct way to express very large or very small numbers. Key exponent rules include the multiplication of like bases, which involves adding the exponents: \( a^m \times a^n = a^{m+n} \).
This can also be applied in reverse for division: \( a^m / a^n = a^{m-n} \). Understanding these rules can significantly aid in solving complex equations involving powers and roots, by converting them into simpler terms through exponentiation.

Logarithms

Logarithms are the inverse operations of exponents. While exponents tell us how many times to multiply a number by itself, logarithms tell us the power needed to get a specific value. The natural logarithm, denoted \( \ln \), specifically uses the constant \( e \) as its base.
This means that in \( \ln e^x \), we are asking, "To what power must \( e \) be raised to yield \( e^x \)?" The answer simplifies directly to \( x \) because \( e \) raised to the power of \( x \) is simply \( e^x \).
Logarithms are essential in many areas of mathematics and science, as they allow us to solve equations where the unknowns are exponents, making them invaluable for calculating compound interest, population growth rates, and more.

Constant e

The constant \( e \) is a mathematical concept that holds great significance in calculus and exponential growth processes. It is approximately equal to 2.71828 and is the base of the natural logarithm. Unlike arbitrary numbers, \( e \) arises naturally in many different mathematical contexts.
Common occurrences include calculations involving continuous growth or compounding, such as finance or population models. In these contexts, \( e \) provides a more accurate representation of growth than simple fixed-rate models.
Because of its unique properties, \( e \) appears in various mathematical formulas, including Euler's identity, which links five fundamental mathematical constants: the number one, zero, \( e \), the imaginary unit \( i \), and \( \pi \). This identity showcases the elegance and interconnected nature of mathematics. Understanding \( e \) is crucial for anyone studying advanced mathematics, as it frequently appears across various topics.