Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the common logarithmic expression without using a calculator. \(\log (10,000)\)

Step-by-Step Solution

Answer

The value of \( \log(10,000) \) is 4.

Step 1: Recall the definition of a common logarithm

A common logarithm is a logarithm with base 10, written as \( \log(x) \), which is the power to which the base 10 must be raised to produce the number \( x \). Thus, \( \log(10,000) \) asks what power of 10 gives us 10,000.

Step 2: Express the number in exponential form

Express 10,000 as a power of 10. Notice that 10,000 can be rewritten as \( 10^4 \), because multiplying 10 by itself 4 times gives 10,000 (i.e., \( 10 \times 10 \times 10 \times 10 = 10,000 \)).

Step 3: Evaluate the logarithm

Using the property of logarithms \( \log(10^n) = n \), substitute 10,000 with \( 10^4 \). Therefore, \( \log(10,000) = \log(10^4) = 4 \).

Key Concepts

Base 10

A common logarithm uses base 10. This is why it's often seen as \( \log(x) \) instead of writing it with an explicit base. Understanding base 10 is crucial because it's the foundation for our number system. It represents powers of ten that we use every day.For example:

• \( 10^0 = 1 \)
• \( 10^1 = 10 \)
• \( 10^2 = 100 \)

Whenever you see \( \log(x) \), it means "what power of 10 gives x?". This is why in our exercise, finding \( \log(10,000) \) means asking what power ten should be raised to, to result in 10,000.Understanding base 10 logarithms can help simplify many problems in mathematics, especially when dealing with large numbers. The base 10 system is also known as the decimal system, used universally in everyday counting.

Exponential Form

Exponential form is a way of representing numbers to make them easier to read and understand, especially when dealing with large quantities. It's written as a number raised to the power of another number. For example, \( 10,000 \) can be converted to \( 10^4 \).So how do we get from \( 10,000 \) to \( 10^4 \)?

• Start by recognizing repeated multiplication in the number.
• \( 10^4 = 10 \times 10 \times 10 \times 10 \) which equals 10,000.

Expressing numbers in this way is extremely useful for simplifying calculations. When we see \( \log(10,000) \), our ability to rewrite it as \( \log(10^4) \) makes solving the problem simpler. Since you know it's base 10, recognize that the exponent is the same as the answer to the logarithm, which in this case is 4.

Logarithmic Properties

Logarithmic properties are rules that apply to simplify and calculate logarithms. These properties are particularly helpful when dealing with logarithms with base 10.One important property is:

• \( \log(10^n) = n \) which means if you can express the number as a power of 10, the power is the answer to the logarithm.

In our exercise, because \( 10,000 \) is \( 10^4 \), the common logarithm \( \log(10,000) \) simplifies directly to 4.Another useful property is the product rule, \( \log(ab) = \log(a) + \log(b) \), and the quotient rule, \( \log(\frac{a}{b}) = \log(a) - \log(b) \). These rules simplify expressions by breaking them down further. Understanding these properties can greatly help in solving complex logarithmic problems, making them fundamental to mastering logarithms.