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Chapter 10: Problem 84

Evaluate each logarithm. $$ \log 10,000 $$

Short Answer

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Step by step solution

01

Understand the Properties of Logarithms

When the base of the logarithm is not specified, we can assume it's 10 (it becomes a common logarithm). So, \(\log 10,000\) is asking us the question: To what power must we raise 10 to obtain 10,000?
02

Solve the Logarithm

10 to the power of what equals 10,000? Since 10,000 is 10 raised to the fourth power (since 10 * 10 * 10 * 10 = 10,000), the answer is 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Logarithms
Common logarithms are a specific type of logarithm that use 10 as their base. Whenever you see a logarithm written as \( \log \) without a base shown, it's a common logarithm, meaning the base is automatically assumed to be 10. This can be very helpful because it simplifies many problems. Instead of worrying about different bases, you just focus on powers of 10. For instance, the expression \( \log 10,000 \) is a perfect example of a common logarithm. The question is: "How many times do you need to multiply 10 by itself to get 10,000?" The answer here is quite straightforward because it's 10 raised to the power of 4.
Power of Ten
A power of ten refers to any number that is a result of multiplying 10 by itself a certain number of times. In the expression \( 10^n \), the 'n' represents how many times the number 10 is multiplied by itself.

Here's an easy breakdown:
  • \( 10^1 = 10 \)
  • \( 10^2 = 100 \)
  • \( 10^3 = 1,000 \)
  • \( 10^4 = 10,000 \)

In the case of \( \log 10,000 \), the solution requires determining the power to which 10 must be raised to equal 10,000. Since \( 10^4 = 10,000 \), the answer is 4. Understanding powers of ten makes it easier to work with common logarithms, as it鈥檚 just a matter of counting how many times you multiply 10.
Logarithmic Properties
Logarithmic properties are tools that help solve logarithmic problems more efficiently. These properties can simplify expressions and make complex calculations much easier. The most basic property of logarithms is the relationship between a logarithm and exponents:

  • Given a base 'b', the expression \( \log_b(b^n) = n \) , which means that if you take the logarithm of a base raised to a power, it equals the exponent.
  • The product property states \( \log_b(MN) = \log_b(M) + \log_b(N) \), which allows you to split a logarithm of a product into a sum of two logs.
  • The quotient property claims \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \), helpful for dividing terms inside a log.
  • Finally, the power property indicates \( \log_b(M^n) = n \cdot \log_b(M) \), meaning you can bring an exponent outside of the log as a multiplier.

These properties can be crucial when solving or simplifying whether it鈥檚 a math homework or a real-world application. For instance, realizing that \( 10,000 \) is just \( 10^4 \) can immediately help you see that \( \log 10,000 \) is 4, harnessing the inherent property of powers in logs.

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