Problem 1 - Contemporary Precalculus

Question

Find the logarithm, without using a calculator. $$\log 10,000$$

Step-by-Step Solution

Answer

Answer: 4

Step 1: Identify the base of the logarithm.

The logarithm is given as \(\log 10,000\). By default, if the base is not specified, it is assumed to be 10. So, the base of the logarithm is 10.

Step 2: Rewrite the number 10,000 as a power of the base.

We need to express 10,000 as a power of the base, which is 10. In order to do that, we write \(10,000\) as \(10^4\).

Step 3: Apply the power rule of logarithms.

According to the power rule of logarithms, we have: $$\log_a (a^x) = x$$ In this case, our base (\(a\)) is 10, and our exponent (\(x\)) is 4. So, we get: $$\log_{10} (10^4) = 4$$

Step 4: Write down the final answer.

Therefore, the value of the logarithm without using a calculator is: $$\log 10,000 = 4$$

Key Concepts

Power Rule of Logarithms

When working with logarithms, one useful rule to understand is the power rule. This rule simplifies the process of finding logarithms, especially for large numbers expressed as powers of a base. The power rule states:

• \( \log_a (a^x) = x \)

What this means is that if you have a logarithm where the argument is a power of its base, the logarithm evaluates to the exponent. For example, if you're dealing with \( \log_{10} (10^4) \), the base here is 10, and the expression \( 10^4 \) means 10 raised to the power of 4. By applying the power rule, \( \log_{10} (10^4) \) simplifies directly to 4 because the exponent is 4.
This rule is a shortcut that really helps when you're calculating logarithms by hand or understanding logarithmic expressions better.

Base of Logarithms

In logarithms, the base is the number that is raised to a power to get the argument. Understanding the base is crucial as it defines the logarithmic scale. In our exercise, we have:

• \( \log 10,000 \)

The base of this logarithm is 10. Generally, if no base is specified in a logarithm, it is assumed to be 10, commonly referred to as the "common logarithm." Knowing this default can save time and avoid confusion. The base tells us how logarithms convert multiplication into addition. For instance, using our base 10 example, 10 is multiplied to itself 4 times to reach 10,000. This idea underlies why the logarithm \( \log_{10} 10,000 \) equals 4, because 10 raised to 4 gives 10,000.

Expressing Numbers as Powers

To find a logarithm manually, often the key step is expressing the number as a power of its base. This involves finding the exponent that transforms the base into the given number. For example, to compute \( \log 10,000 \), where the base is 10, we express 10,000 as an exponential power:

• \( 10,000 = 10^4 \)

This process aligns with identifying how many times the base needs to be multiplied by itself to achieve the target number. Recognizing this pattern enables you to quickly use the power rule of logarithms. If the base and argument are correctly aligned, expressing numbers as powers cuts down problem-solving time and aids in intuitive understanding. By recognizing these patterns, one can deftly maneuver through varied logarithmic computations without the use of a calculator.