Problem 92 - College Algebra

Question

Evaluate or simplify each expression without using a calculator. $$ \ln \frac{1}{e^{7}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is -7

Step 1: Apply the Logarithmic Property of Exponentiation

Using the property of logarithms that states that log_b(a^n) = n*log_b(a), the given expression could be rewritten by bringing down the exponent as \( -7 \times \ln e \)

Step 2: Use the Property of Natural Logarithms

Given that the base of the natural logarithm is 'e', it holds that \( \ln e = 1 \). Substituting \( \ln e \) in the expression from Step 1, we get \( -7 \times 1 \))

Step 3: Evaluate the Expression

After substituting \( \ln e \) with '1' , we perform the operation to get the final result. Thus (-7 * 1) equals -7

Key Concepts

Logarithmic Property of Exponentiation

Understanding the logarithmic property of exponentiation is essential to simplify expressions involving logarithms and exponents. When you see an expression like the natural logarithm of a number raised to a power, such as \( \text{ln} \frac{1}{e^{7}} \), the exponentiation property can be used to simplify the calculation.

This property essentially says that the log of a power can be expressed as the exponent times the log of the base. Mathematically, it is presented as: \( \text{log}_b (a^n) = n \times \text{log}_b (a) \). Applying this, you can move the exponent outside of the logarithm, which in our example would result in \( -7 \times \text{ln}(e) \).

Using this simplification step makes handling logarithmic expressions more manageable, and it is particularly useful when dealing with natural logarithms, where the base of the log is the mathematical constant \( e \).

Natural Logarithms

Natural logarithms are a type of logarithm where the base is the mathematical constant \( e \), approximately equal to 2.718. In notation, a natural logarithm is represented by \( \text{ln} \). One of the fundamental properties of natural logarithms is that the natural logarithm of \( e \) is 1, that is, \( \text{ln}(e) = 1 \).

When simplifying expressions involving natural logarithms, knowing this property is invaluable. For instance, in the exercise \( \text{ln} \frac{1}{e^{7}} \), we first use the logarithmic property of exponentiation and then apply the fact that \( \text{ln}(e) = 1 \) to simplify the expression further.

Example:

If you encounter \( \text{ln}(e^5) \), you can immediately know that the simplified form is 5 because \( \text{ln}(e) = 1 \). This makes natural logarithms particularly straightforward when the expressions are powers of \( e \).

Simplifying Expressions

Simplifying expressions in mathematics is much like decluttering a room; it's all about making complex, messy situations clearer and more manageable. When you're faced with a complicated expression, especially those involving logarithms and exponents, your goal is to reduce it to its simplest form.

To simplify effectively, you need to understand and apply various mathematical properties and rules. For logarithms and natural logarithms, the properties we've discussed help transform the expressions into something much simpler.

Step-by-Step Approach:

• Identify the parts of the expression you can simplify using known properties.
• Apply these properties, such as converting the natural log of a power to a multiple, or recognizing that \( \text{ln}(e) \) equals 1.
• Perform any arithmetic operations left to find the simplest form of the expression.

Continuing with our example, after applying the logarithmic property of exponentiation and the defining property of natural logarithms, we are left with an arithmetic operation of \( -7 \times 1 \), giving us the final simplified result of -7.