Problem 18 - Calculus of a Single Variable

Question

Using Properties of Logarithms In Exercises 17 and \(18,\) use the properties of logarithms to approximate the indicated logarithms, given that \(\ln 2 \approx 0.6931\) and \(\ln 3 \approx 1.0986\) \(\begin{array}{llll}{\text { (a) } \ln 0.25} & {\text { (b) } \ln 24} & {\text { (c) } \ln \sqrt[3]{12}} & {\text { (d) } \ln \frac{1}{72}}\end{array}\)

Step-by-Step Solution

Answer

\((a) -1.3862 \), \((b) 3.1783 \), \((c) 0.87036 \), \((d) -4.15888 \)

Step 1: Solve \( \ln 0.25 \)

First, rewrite \( 0.25 \) as \( \frac{1}{4} \) which also equals \( 2^{-2} \). Apply the power rule of logarithms which allows \( \ln 2^{-2} \) to be rewritten as \( -2 \ln 2 \). Using the provided \( \ln 2 = 0.6931 \), multiply \( -2 \) by this value to get \( -1.3862 \).

Step 2: Solve \( \ln 24 \)

First, rewrite \( 24 \) as the product of powers of 2 and 3, i.e., \( 24 = 2^3 \times 3 = 8 \times 3 \). Apply the rule of logs that states that the log of a product equals the sum of the logs, i.e., \( \ln 24 = \ln (2^3 \times 3) = \ln 2^3 + \ln 3 \). Use the power rule of logs to rewrite \( \ln 2^3 \) as \( 3 \ln 2 \), and add it to \( \ln 3 \). Substitute the provided values \( \ln 2 = 0.6931 \) and \( \ln 3 = 1.0986 \) to get \( 3.1783 \).

Step 3: Solve \( \ln \sqrt[3]{12} \)

Convert cube root to power form, \( \sqrt[3]{12} \) becomes \( 12^{1/3} \). Now rewrite 12 as \( 2^2 \times 3 \). So, \( \ln \sqrt[3]{12} = \ln (2^2 \times 3)^{1/3} = \frac{1}{3}(2\ln2 + \ln3) \). Applying provided values of \( \ln 2 = 0.6931 \) and \( \ln 3 = 1.0986 \), the result will be \( 0.87036 \).

Step 4: Solve \( \ln \frac{1}{72} \)

First, rewrite \( \frac{1}{72} \) as \( 2^{-3} \times 3^{-2} \). Then, applying the property that the log of a quotient equals the difference of logs, \( \ln \frac{1}{72} = -3 \ln 2 - 2 \ln 3 \). Plugging in the given approximations \( \ln 2 = 0.6931 \) and \( \ln 3 = 1.0986 \), yields \( -4.15888 \).

Key Concepts

Properties of Logarithms

Logarithms have several key properties that make them incredibly useful in simplifying mathematical expressions, especially when working with multiplication, division, or exponentiation. These properties include:

• Product Rule: The logarithm of a product is the sum of the logarithms: \( \log_b(xy) = \log_b(x) + \log_b(y) \).
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \).
• Power Rule: The logarithm of a power is the exponent times the logarithm of the base: \( \log_b(x^n) = n \cdot \log_b(x) \).

These properties allow us to break down complex logarithmic expressions into simpler parts, which can be very helpful for computation and solving equations.

Natural Logarithms

A natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). Natural logarithms are often used in calculus and exponential growth problems because the number \( e \) is a fundamental constant in mathematics.

• Why Use Natural Logarithms? They simplify calculations involving growth rates and decay such as compound interest, population growth, and radioactive decay.
• Notation: \( \ln(x) = \log_e(x) \), where \( \ln \) is more common in mathematical and scientific contexts.

Natural logarithms have the same properties as other logarithms; you can apply product, quotient, and power rules to them, just like any other logarithmic function.

Logarithmic Functions

Logarithmic functions are the inverse operations of exponential functions. If an exponential function is represented as \( y = b^x \), then its inverse is \( x = \log_b(y) \).

• Graph Characteristics: Logarithmic graphs have a characteristic curve that increases slowly and never touches the x-axis, known as an asymptote.
• Domain and Range: The domain of a logarithmic function is all positive real numbers, and the range is all real numbers.

Understanding these functions helps solve various equations and model real-world phenomena, such as sound intensity (decibels) and pH levels.

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another, which is useful if your calculator only supports logarithms of certain bases, such as 10 or \( e \).

The change of base formula is given by:
\[ \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \]
This formula lets you calculate any logarithm using a different base. For example, to compute \( \log_2(8) \) using the base 10 logarithm, use:
\[ \log_2(8) = \frac{\log_{10}(8)}{\log_{10}(2)} \]
By using this formula, you can solve logarithmic expressions without being restricted to the available functions on your calculator.