Problem 23 - Algebra and Trigonometry

Question

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{8} 0.25} & {\text { (b) } \ln e^{4}} & {\text { (c) } \ln (1 / e)} & {}\end{array} $$

Step-by-Step Solution

Answer

(a) \(-\frac{2}{3}\), (b) 4, (c) \(-1\)

Step 1: Evaluate \( \log_{8} 0.25 \)

To evaluate \( \log_{8} 0.25 \), express 0.25 as a power of 8. Since \( 0.25 = \frac{1}{4} \) and knowing \( 8 = 2^3 \), we find that \( 0.25 = \frac{1}{2^2} = 2^{-2} \). Now we need to express it in base 8: \( 2^{-2} = (2^3)^{-2/3} = 8^{-2/3} \). Therefore, \( \log_{8} 0.25 = -\frac{2}{3} \).

Step 2: Evaluate \( \ln e^{4} \)

Using the property of logarithms \( \ln e^x = x \), we apply this property here. Since \( x = 4 \), therefore \( \ln e^{4} = 4 \).

Step 3: Evaluate \( \ln (1 / e) \)

Using the property \( \ln (\frac{1}{a}) = - \ln a \), substitute \( a = e \). Therefore, \( \ln (\frac{1}{e}) = - \ln e = -1 \).

Key Concepts

Logarithmic Properties

Logarithmic properties are essential for simplifying complex logarithmic expressions. These properties include product, quotient, and power rules, allowing us to rewrite and solve logarithms more efficiently. Here's a brief overview of some core properties:

• **Product Rule:** For two positive numbers, the logarithm of their product is the sum of their logarithms: \( \log_b (MN) = \log_b M + \log_b N \).


• **Quotient Rule:** For division, the logarithm of the quotient is the difference of their logarithms: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \).


• **Power Rule:** When raising a power, the logarithm of a number raised to an exponent is the exponent times the logarithm of the number: \( \log_b (M^p) = p \cdot \log_b M \).

These properties help streamline calculations and are extremely useful in various mathematical and scientific applications.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.718. It is widely used in calculus and real-world applications due to its natural occurrence in growth processes. Here are key points about natural logarithms:

• **Basic Property:**\( \ln e = 1 \)


• **Exponent Rule:**For a natural logarithm of an exponential expression, such as \( \ln(e^x) \), the solution simplifies to the exponent itself: \( \ln(e^x) = x \).


• **Quotient Rule:**Similarly, \( \ln(\frac{1}{e}) = -1 \) by applying the property of reciprocals \( \ln(\frac{1}{a}) = -\ln(a) \).

These rules make calculations involving natural logarithms simpler and more intuitive.

Exponential Functions

Exponential functions are mathematical functions of the form \( f(x) = a \cdot b^x \) where \( b \) is a positive real number not equal to one. They model exponential growth or decay processes, which are common in areas like population dynamics, finance, and physics. Key characteristics include:

• **Base \( e \):**A unique base where exponential functions are often expressed due to the natural properties of \( e \), simplifying differentiation and integration in calculus.


• **Exponential Growth and Decay:**If \( b > 1 \), the function exhibits growth. If \( 0 < b < 1 \), it exhibits decay. Real-world cases can include compound interest calculation and radioactive decay.


• **Relationship with Logarithms:**Inversely related; while logarithms deal with exponents, exponential functions reject exponents to find the original number.

Understanding these concepts is crucial as they apply to a multitude of scientific fields and everyday scenarios.