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Chapter 12: Problem 18

Evaluate each logarithm to four decimal places. \(\log \left(2.13 \times 10^{4}\right)\)

Short Answer

Expert verified
\( 4.3284 \)

Step by step solution

01

- Use Logarithmic Property

Recall that \ \ \ \ \ \ \ \ \( \log(ab) = \log(a) + \log(b) \). In this expression, let \( a = 2.13 \ \text{and} \ b = 10^{4} \). Then rewrite the logarithm: \ \ \( \log(2.13 \times 10^4) = \log(2.13) + \log(10^4) \).
02

- Evaluate \( \log(10^4) \)

Remember that the logarithm of a power of 10 is just the exponent. Therefore, \ \( \log(10^4) = 4 \).
03

- Evaluate \( \log(2.13) \)

Using a scientific calculator, evaluate \( \log(2.13) \). The result is approximately 0.3284.
04

- Add the Two Values

Add the two logarithmic values obtained in Step 2 and Step 3: \ \( \log(2.13 \times 10^4) = 0.3284 + 4 = 4.3284 \).

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Key Concepts

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

logarithmic properties
Understanding logarithmic properties is crucial when solving logarithm problems. These properties simplify complex logarithmic expressions by breaking them down into smaller parts. One key property is the product rule for logarithms, which states that:
\ \( \ \log(ab) = \ \log(a) + \ \log(b) \ \)
This rule helps transform products of numbers inside the logarithm into sums of logarithms, making calculations easier.

In our exercise, we applied this property by breaking down \ \( \ \log(2.13 \ \times 10^4) \ \) into \ \( \ \log(2.13) + \ \log(10^4) \ \). This separation allows us to handle each part individually and then combine the results.
scientific calculator
A scientific calculator is an indispensable tool when dealing with logarithms, especially for non-integer values. These calculators have a dedicated \ \(\ \log \ \) button that finds the base-10 logarithm of any given number.

To evaluate \ \( \ \log(2.13) \ \), simply enter the value into the calculator and press the \ \( \ \log \ \) button. The display should show approximately 0.3284, as we found in the exercise.

Using a calculator ensures accuracy and speeds up the process, especially when dealing with more complex or larger numbers.
logarithm of a power of 10
Logarithms of powers of 10 are particularly straightforward. The logarithm of 10 raised to any power is simply the exponent itself. This is derived from the definition of logarithms:

\ \( \ \log(10^n) = n \ \)

So, in our example, \ \( \ \log(10^4) = 4 \ \). This rule makes calculations involving powers of 10 very simple and quick.

Understanding this property allows you to instantly identify the logarithmic value of any power of 10, removing the need for lengthy calculations or even a calculator for this part.

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Most popular questions from this chapter

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