Problem 3 - Thomas Calculus

Question

Use the properties of logarithms to simplify the expressions in Exercises 3 and \(4 .\) $$ \begin{array}{l}{\text { a. } \ln \sin \theta-\ln \left(\frac{\sin \theta}{5}\right) \quad \text { b. } \ln \left(3 x^{2}-9 x\right)+\ln \left(\frac{1}{3 x}\right)} \\ {\text { c. } \frac{1}{2} \ln \left(4 t^{4}\right)-\ln 2}\end{array} $$

Step-by-Step Solution

Answer

a. \( \ln 5 \), b. \( \ln (x-3) \), c. \( \ln (t^2) \)

Step 1: Simplify Expression a

For expression a, we have \[ \ln \sin \theta - \ln \left(\frac{\sin \theta}{5}\right) \]Using the property of logarithms that states \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \), we can rewrite the expression as:\[ \ln \left( \frac{\sin \theta}{\frac{\sin \theta}{5}} \right) = \ln (5) \]

Step 2: Simplify Expression b

For expression b, we have\[ \ln \left( 3x^2 - 9x \right) + \ln \left( \frac{1}{3x} \right) \] First, apply the property \( \ln a + \ln b = \ln (a \cdot b) \):\[ \ln \left( \left(3x^2 - 9x\right) \cdot \frac{1}{3x} \right) \]Simplify the expression inside the logarithm. Factor \(3x\) from \(3x^2 - 9x\) to get \(3x(x - 3)\). The expression simplifies to:\[ \ln \left( (x-3) \cdot \frac{1}{1} \right) = \ln (x-3) \]

Step 3: Simplify Expression c

For expression c, we have\[ \frac{1}{2} \ln (4t^4) - \ln 2 \]First, apply the power rule for logarithms: \( \frac{1}{2} \ln (4t^4) = \ln \left((4t^4)^{1/2}\right) = \ln (2t^2) \).Now, use the property \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \):\[ \ln \left( \frac{2t^2}{2} \right) = \ln (t^2) \]

Step 4: Compile Final Simplified Expressions

Let's compile the results from each expression: - Expression a simplifies to: \( \ln 5 \)- Expression b simplifies to: \( \ln (x-3) \)- Expression c simplifies to: \( \ln (t^2) \)

Key Concepts

Simplifying Logarithmic Expressions

When working with logarithms, simplifying expressions can make complex functions more understandable. The properties of logarithms are extremely useful in breaking down and combining logarithmic forms systematically.

Some key properties used in simplification include:

• **Product Rule**: \( \ln a + \ln b = \ln(ab) \) - You combine two logarithms into one by multiplying their respective arguments.
• **Quotient Rule**: \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \) - This is used to combine two logarithms by dividing their arguments, resulting in a simpler form.
• **Power Rule**: \( a\ln b = \ln(b^a) \) - Allows you to move a coefficient in front of a logarithm into the argument of the logarithm, raising it to a power.

Knowing these core rules allows you to manipulate and simplify logarithmic expressions easily.

For example, consider simplifying \( \ln \sin \theta - \ln \left(\frac{\sin \theta}{5}\right) \). By applying the quotient rule, you can rewrite this expression as \( \ln 5 \). This demonstrates the effectiveness of using logarithmic properties to simplify complex expressions into straightforward ones.

Logarithmic Identities

Logarithmic identities are rules that help in equating and simplifying logarithmic terms. They form the backbone of understanding and solving logarithmic equations.

Some vital identities include:

• **Identity Logarithm**: \( \ln 1 = 0 \) and \( \ln e = 1 \) - Useful when recognizing or simplifying logarithms of base values.
• **Inverse Properties**: \( e^{\ln x} = x \) and \( \ln(e^x) = x \) - Highlight the inverse relationship between exponentials and logarithms.
• **Change of Base Formula**: While not used directly in simplification, this allows calculation of logarithms with different bases: \( \log_b x = \frac{\ln x}{\ln b} \).

These identities show that logarithms turn multiplication, division, and exponentiation into their simpler arithmetic forms: addition, subtraction, and multiplication.

In examples like \( \ln((3x^2 - 9x) \cdot \frac{1}{3x}) \), recognizing and applying these identities allows for simplification to \( \ln(x - 3) \). Observing patterns and using identities can streamline solving more intricate logarithmic problems.

Logarithm Rules in Calculus

Logarithmic functions appear frequently in calculus, often when differentiating or integrating functions. Understanding how to use logarithm rules can provide clarity and ease in solving calculus problems.

In calculus, you often use logarithm rules to simplify differentiation and integration:

• **Differentiation**: The derivative of \( \ln x \) is \( \frac{1}{x} \). This simplicity aids in solving more complex functions where a logarithm is involved, as seen in deriving composite functions.
• **Integration**: The integral of \( \frac{1}{x} \) is \( \ln |x| + C \). This rule helps convert complex fraction integrals into logarithmic forms.

By using the logarithm rules such as the power rule, you can also simplify logarithmic expressions before differentiation. For instance, transforming \( \frac{1}{2} \ln (4t^4) - \ln 2 \) into \( \ln (t^2) \) by applying the power and quotient rules simplifies the task of finding derivatives or integrals. Thus, familiarizing with logarithm rules not only aids in algebra but extends to calculus for smoother computation.