Problem 15 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression without using a calculator. $$\log 10,000$$

Step-by-Step Solution

Answer

\(\log_{10} 10,000 = 4\)

Step 1: Understanding the exercise

The equation given is \(\log_{10} 10,000\). We need to figure out what power or exponent 10 must be raised to in order to get 10,000.

Step 1: Break down the number

The number 10,000 can be written as \(10^{4}\). This is because \(10*10*10*10 = 10,000\). So, our equation now looks like this: \(\log_{10} 10^{4}\)

Step 2: Apply the Logarithm Rule

We can now apply the logarithm rule \(\log_{a} a^n = n\). We match \(a = 10\), \(n = 4\), and according to the rule mentioned, \(\log_{10} 10^{4} = 4\).

Step 4: Final Answer

Therefore, \(\log_{10} 10,000 = 4\)

Key Concepts

Understanding Exponents

Exponents are a crucial concept in mathematics, indicating how many times a number, known as the base, is multiplied by itself. For example, in the expression \(10^4\), the base is 10, and the exponent is 4. This indicates that you multiply 10 by itself four times: \(10 \times 10 \times 10 \times 10\), which equals 10,000.

Exponents simplify the way we write and understand repeated multiplication. For instance:

• \(2^3\) is the same as \(2 \times 2 \times 2 = 8\).
• \(5^2\) implies \(5 \times 5 = 25\).

Breaking down numbers using exponents can make complex multiplications easier to handle and understand, especially when dealing with powers of base 10.

Base 10 in Mathematics

Base 10, also known as the decimal system, is the foundation of our number system. The number 10 serves not only as a count, but also forms the basis of logarithms and the base for calculating powers in many scenarios.

When dealing with logarithms, the base 10 is often implied and can be written as \(\log_{10}\) or simply \(\log\). Using base 10 is especially helpful when evaluating expressions like \(\log 10,000\), as it directly relates to how many times you can multiply 10 by itself to reach the number 10,000.

For example, the base transformation for 10,000 to \(10^4\) translates directly into the logarithmic expression \(\log_{10} 10,000 = 4\). Understanding base 10 helps in simplifying mathematical operations and grasping core concepts of powers and roots.

Logarithm Rules Explained

Logarithms are the inverse operation of exponents. They determine the power to which a base number must be raised to obtain a given value. The rule \(\log_a a^n = n\) is fundamental when solving logarithmic expressions.

This rule implies that if you have a base \(a\) raised to the power \(n\), the logarithm of that expression with the same base \(a\) will simply return \(n\).

For instance, in our exercise:

• The expression \(\log_{10} 10,000\) was rewritten as \(\log_{10} 10^4\), thanks to the exponential property.
• By applying the rule \(\log_{a} a^n = n\), we deduce that \(\log_{10} 10^4 = 4\).

This rule is pivotal because it simplifies the process of finding logarithms without extensive calculations, making it easier to understand and compute large numbers or powers.