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Chapter 13: Problem 4

Use your calculator to find the log of the following numbers. (a) \(10^{4}\) (b) \(1 \times 10^{-6}\) (c) \(1.7 \times 10^{8}\) (d) \(10^{-8}\) (e) 10

Short Answer

Expert verified
The logs of the given numbers are: (a) \(10^{4}\) is 4, (b) \(1 \times 10^{-6}\) is -6, (c) \(1.7 \times 10^{8}\) is approximately 8.23, (d) \(10^{-8}\) is -8 and (e) 10 is 1.

Step by step solution

01

Calculating the log of \(10^{4}\)

Use your calculator to calculate the logarithm to the base 10 of \(10^{4}\), which is simply 4 by the rule of logarithm for exponents (\( \log_b {b^n} = n )\).
02

Calculating the log of \(1 \times 10^{-6}\)

Use your calculator to find the log to the base 10 of \(1 \times 10^{-6}\). By the rule of logarithm for exponents, you can separate the 1 and \(10^{-6}\) as separate components and find their logs, which yields \( \log{1} + \log{10^{-6}} \). Logarithm of 1 to any base is 0 by rule, and logarithm of \(10^{-6}\) to base 10 is -6 by rule, so the answer is -6.
03

Calculating the log of \(1.7 \times 10^{8}\)

Use your calculator to find the log to the base 10 of \(1.7 \times 10^{8}\). By the rule of logarithm for products, you can separate the 1.7 and \(10^{8}\) as separate components and find their logs, which is \( \log{1.7} + \log{10^{8}} \). Using a calculator, \( \log{1.7} \) approximates 0.23. Logarithm of \(10^{8}\) to base 10 is 8 by rule, so the answer is approximately 8.23.
04

Calculating the log of \(10^{-8}\)

Use your calculator to find the log to the base 10 of \(10^{-8}\), which is simply -8 by the rule of logarithm for exponents .
05

Calculating the log of 10

Use your calculator to find the log to the base 10 of 10, which is simply 1 by the rule of logarithm for exponents .

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Key Concepts

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

Logarithm Rules
Understanding logarithm rules is foundational for solving logarithm problems. A logarithm, in simplest terms, is the inverse operation to exponentiation. The notation \( \log_b(x) \) is read as "log base \( b \) of \( x \)". There are several key rules that simplify understanding and calculating logarithms.
  • The product rule: \( \log_b{(xy)} = \log_b{(x)} + \log_b{(y)} \). This states that the log of a product is the sum of the logs.
  • The quotient rule: \( \log_b{\left(\frac{x}{y}\right)} = \log_b{(x)} - \log_b{(y)} \). This expresses the log of a quotient as the difference of the logs.
  • The power rule: \( \log_b{(x^n)} = n \log_b{(x)} \). This shows that the log of a power is the exponent times the log.
  • The change of base formula: \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \). This allows you to switch the base of the logarithm.

Additionally, special cases include: \( \log_b(b) = 1 \) because \( b^1 = b \), and \( \log_b(1) = 0 \) because \( b^0 = 1 \). These rules are essential for breaking down complex logarithmic expressions.
Logarithm Calculations
Calculating logarithms can seem tricky at first, but with practice, it becomes easier.
When approaching a logarithm problem, first identify if you can apply any basic rules. In the examples given in step-by-step solutions, numbers with a base of 10 offer direct uses of these rules.
  • For instance, \( \log_{10}(10^4) \) simplifies directly to 4 using the power rule: \( \log_b{b^n} = n \).
  • Another example is \( \log_{10}(10^{-6}) \), which simplifies to -6, again using the power rule.

Sometimes, you may need a calculator, like when dealing with non-integers or other bases.
For \( \log_{10}(1.7 \times 10^8) \), you should separate it into \( \log_{10}(1.7) + \log_{10}(10^8) \), calculating each part separately.
  • Typically, the first part, \( \log_{10}(1.7) \), needs a calculator, while the second part, \( \log_{10}(10^8) \), simplifies to 8.

Remember to use logarithm rules as shortcuts, confirming that each calculation obeys the mathematical laws of logarithms.
Exponents and Logarithms
Exponents and logarithms share a close relationship because logarithms are essentially inverted exponent operations.
Understanding this relationship deepens comprehension of both concepts.
  • Exponential form shows a number being multiplied by itself: e.g., \( b^n \) means \( b \) multiplied \( n \) times.
  • Logarithmic form answers the question: "To what power must we raise the base \( b \) to obtain x?"

By understanding this inverse relationship, it is easier to see how equations involving exponents can be converted into logarithmic form to simplify solutions.
The rule \( b^{\log_b(x)} = x \) demonstrates the interplay between the two; any number raised to the logarithm of the same base returns the original number.
  • Also, \( \log_b(b^x) = x \) confirms that logarithms are the exponents needed to achieve the number with that base.
  • Using these forms interchangeably makes it easier to solve problems involving both exponents and logarithms.

This understanding is crucial since many real-world phenomena use exponential and logarithmic models, from computing interest rates to measuring earthquake magnitudes.

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