Problem 38 - College Algebra

Question

For the following exercises, use the definition of common and natural logarithms to simplify. $$ 2 \log (0.0001) $$

Step-by-Step Solution

Answer

The simplified expression is -8.

Step 1: Understand the Problem

You are given the expression \( 2 \log(0.0001) \). The task is to simplify this expression using your understanding of logarithms.

Step 2: Recall the Definition of Logarithm

The logarithm \( \log(a) \) refers to the power to which the base, usually 10 for common logarithms, must be raised to produce \( a \). Thus, if \( \log(b) = c \), then \( 10^c = b \).

Step 3: Rewrite 0.0001 in Exponential Form

Rewrite the number 0.0001 as a power of 10. We know that \( 0.0001 = 10^{-4} \).

Step 4: Apply Logarithmic Rule

Use the property of logarithms that \( \log(10^c) = c \). Therefore, \( \log(0.0001) = \log(10^{-4}) = -4 \).

Step 5: Multiply by 2

Now, substitute \( \log(0.0001) \) with \(-4\): \( 2 \times (-4) \).

Step 6: Simplify the Expression

Calculate \( 2 \times (-4) = -8 \). Thus the simplified expression is \(-8\).

Key Concepts

Common Logarithms

Common logarithms use base 10, which is the number system most familiar to us. When you see "log" without a base highlighted, it typically means the common logarithm, denoted as \( \log_{10} \). This is because the calculations often involve numbers in base 10, making these logarithms particularly handy.

• For example, if you have a logarithm such as \( \log(1000) \), it asks, "what power should 10 be raised to give 1000?" The answer is 3 because \( 10^3 = 1000 \).
• The same applies to smaller numbers; for \( \log(0.0001) \), it is asking for what power of 10 results in 0.0001. Rewritten in exponential form, this is \( 10^{-4} \), which means the logarithm of 0.0001 with base 10 is -4.

By understanding common logarithms in terms of powers of 10, it becomes easier to mentally calculate or estimate log values.

Natural Logarithms

Natural logarithms use the number "e" (approximately 2.71828) as the base. Instead of being indicated by "log", natural logarithms are denoted by "ln". They are widely used in higher mathematics, particularly calculus, because of their natural occurrence in growth processes, such as population growth or interest calculations.

• Applying the concept of logarithms to "e" works similarly. If \( \ln(e^2) \), it is asking for the power to which "e" must be raised to get \( e^2 \). The answer here is simply 2.
• Another example if you need to solve \( \ln(1) \), it follows that any number raised to the power of 0 equals 1, therefore \( \ln(1) = 0 \).

While different from common logarithms, natural logarithms follow the same principle of expressing a number as a power of their respective bases.

Exponential Form

The exponential form is a powerful tool with its roots in expressing numbers as powers of base numbers. Whether working with base 10, "e", or any other numbers, the concept remains simple: convert numbers into a more manageable notation.

• The earlier example of 0.0001 can be effectively expressed in exponential form as \( 10^{-4} \). This representation takes a long decimal and expresses it succinctly.
• Another example with natural logarithms: "e" raised to the power of 3, \( e^3 \), indicates the natural exponent form relevant in many growth models.

Utilizing exponential form helps streamline calculations and supports understanding complex logarithmic problems by presenting them in a simplified format. Efficiently switching between logarithmic and exponential forms is key to mastering these concepts.