Problem 8 - College Algebra with Modeling and Visualization

Question

Evaluate each expression by hand, if possible. (a) \(\log (-4)\) (b) \(\log 1\) (c) \(\log 0\) (d) \(-6 \log 100\)

Step-by-Step Solution

Answer

(a) Undefined, (b) 0, (c) Undefined, (d) -12.

Step 1: Understanding Logarithms and Negative Numbers

The logarithm of a number represents the power to which the base must be raised to obtain that number. It's important to note that, in real numbers, you cannot take the logarithm of a negative number. Thus, \( \log(-4) \) is undefined.

Step 2: Evaluate \( \log(1) \)

The logarithm of 1 at any base is always 0. This is because any non-zero number raised to the power of zero equals 1. Therefore, \( \log(1) = 0 \).

Step 3: Evaluate \( \log(0) \)

Logarithm of 0 is undefined in real numbers. This is because there is no exponent you can raise a non-zero base to that will result in zero.

Step 4: Calculate \( -6 \log(100) \)

Using the property of logarithms that states \( \log_{10}(100) = 2 \) (since \( 10^2 = 100 \)), substitute this into the expression: \[ -6 \cdot \log_{10}(100) = -6 \cdot 2 = -12. \] Thus, \( -6 \log(100) = -12 \).

Key Concepts

Undefined Logarithms

Logarithms can often seem puzzling, especially when certain logarithmic expressions are said to be "undefined." This term is used when there's no real value that satisfies the logarithmic equation. One key situation is when you attempt to find the logarithm of a negative number or zero. In real numbers,

• the logarithm of a negative number such as \( \log(-4) \) is undefined because no real number raised to any power gives a negative number.
• Similarly, \( \log(0) \) is undefined, since no positive number raised to any power can yield zero.

Understanding why these logarithms are undefined is vital, as it helps prevent errors in calculations and deepens comprehension of logarithmic behavior.

Logarithm Rules

To work with logarithms effectively, several basic rules are essential. These rules simplify calculations and help solve logarithmic problems. A foundational rule is the logarithm of one, such as \( \log(1) = 0 \). This arises from the fact that any number raised to the power of zero is one.

• Another rule is that the logarithm of a base raised to a power equals the exponent: \( \log_b(b^x) = x \).
• Also, consider the change of base rule, expressed as \( \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \), which facilitates working with different bases.

Practicing these rules will enhance your ability to solve logarithmic expressions efficiently.

Properties of Logarithms

Logarithms possess distinctive properties that are incredibly useful in mathematics. Recognizing these properties allows for the manipulation and simplification of complex expressions. The product, quotient, and power properties are particularly noteworthy:

• Product Property: \( \log_b(MN) = \log_bM + \log_bN \), enabling the breakdown of complex products.
• Quotient Property: \( \log_b\frac{M}{N} = \log_bM - \log_bN \), useful for simplifying ratios.
• Power Property: \( \log_b(M^n) = n \times \log_bM \), critical in linearizing exponential functions.

The application of these properties simplifies calculating logarithms, transforming intricate problems into manageable solutions, such as deriving \( -6 \log(100) = -12 \) using the power property.