Problem 92 - Algebra and Trigonometry

Question

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln \frac{1}{e^{7}} $$

Step-by-Step Solution

Answer

-7

Step 1: Use the properties of logarithms

Apply the property of logarithm that \(\ln \frac{1}{x}\) equals \(-\ln x\). So, you rewrite the given expression as \(-\ln e^{7}\).

Step 2: Apply the power rule of logarithms

The power rule/law of logarithms states that \(\ln a^{n}\) equals \(n \cdot \ln a\). Applying this law to the expression from step 1 results in a simplified expression: \( -7*\ln e\).

Step 3: Value of \(\ln e\)

\(\ln e\) is equal to 1, because the natural logarithm of \(e\) (Euler's number, approximately equal to 2.71828) is 1. This implies that −7 multiplied by 1 equals −7.

Key Concepts

Logarithm Properties

Understanding logarithm properties is crucial in simplifying expressions like \( \ln \frac{1}{e^7} \). One essential property is that \( \ln \frac{1}{x} \) equals \( -\ln x \). This means if you take the natural logarithm of a fraction, you can simply take the negative logarithm of the denominator.

• This property helps you handle fractions within logarithms.
• It's a fundamental tool for rewriting more complex logarithmic expressions.

In the exercise, we used this property to change \( \ln \frac{1}{e^7} \) into \( -\ln e^7 \). This is the starting point for simplifying the expression further.

Power Rule of Logarithms

The power rule of logarithms states that \( \ln a^n \) is equal to \( n \cdot \ln a \). This rule makes it easy to deal with logarithms of exponential expressions by pulling the exponent out in front of the logarithm.

• If you're working with \( \ln e^7 \), the power rule lets you rewrite this as \( 7 \cdot \ln e \).
• This simplification reduces the complexity of the expression.

In our step-by-step solution, we applied this rule after using the properties of logarithms to reach the expression \( -7 \cdot \ln e \). This step is vital for making the final calculation straightforward.

Euler's Number

Euler's number, denoted as \( e \), is approximately 2.71828 and holds a special place in mathematics. It is the base of the natural logarithms, making \( \ln e = 1 \). This means that the natural logarithm of \( e \) simplifies to 1 effortlessly.

• Knowing that \( \ln e = 1 \) is key for calculations involving natural logs.
• It simplifies expressions by reducing terms that multiply with \( \ln e \).

In the exercise, since \( \ln e \) equals 1, the expression \( -7 \cdot \ln e \) simplifies to \( -7 \), showing the utility and simplicity provided by Euler's number.

Logarithm Simplification

Logarithm simplification combines multiple rules and properties to reduce a complex expression to its simplest form.

• Start by identifying all applicable properties and rules, such as those for fractions and exponents.
• Utilize the power rule and properties of \( e \) to reduce terms systematically.

Applying these methods, the given expression \( \ln \frac{1}{e^7} \) simplifies step-by-step:1. Rewrite using fractions property: \( -\ln e^7 \).2. Apply power rule: \( -7 \cdot \ln e \).3. Use the value of \( \ln e \): result is \( -7 \).Each step builds upon the last, showing how fundamental understanding leads to a neat and simple solution.