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Chapter 12: Problem 35

Use a calculator to approximate each logarithm to four decimal places. \(\log _{1 / 4} 12\)

Short Answer

Expert verified
-1.7931

Step by step solution

01

Understanding the Change of Base Formula

To find the value of \( \log_{1 / 4} 12 \) using a calculator, the change of base formula is used. The change of base formula is: \[ \log_b a = \frac{\log_c a}{\log_c b} \] where \( c \) is typically taken as 10 or \( e \) (natural logarithm).
02

Setting Up the Problem

In this exercise, \( b = \frac{1}{4} \) and \( a = 12 \.\) Thus, using the formula, \[ \log_{1 / 4} 12 = \frac{\log 12}{\log (1 / 4)} \].
03

Calculating Logarithms Using a Calculator

Use a calculator to find \( \log 12 \) and \( \log (1 / 4) \): \[ \log 12 \approx 1.0792 \] \[ \log (1 / 4) \approx -0.6021 \].
04

Applying the Change of Base Formula

Plug in the calculated values into the formula: \[ \log_{1 / 4} 12 = \frac{1.0792}{-0.6021} \approx -1.7931 \]

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Key Concepts

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

Logarithms
Logarithms are mathematical operations that determine the power to which a base number must be raised to produce a given number. For example, if you have \( \log_{2} 8 \), you are looking for the power to which 2 must be raised to get 8. The answer is 3 because \( 2^3 = 8 \). This concept is key for understanding the change of base formula. Logarithms can have any base, like 10, 2, or e. They are useful in various fields including science and engineering. Understanding logarithms will help you solve exponential equations and understand growth rates.

A base-10 logarithm is common and often written as \( \log \), while a natural logarithm uses base e and is written as \( \ln \).
Calculator Usage
Using a calculator to solve logarithmic values is straightforward once you understand the function keys. Most scientific calculators have dedicated keys for \( \log \) and \( \ln \). To find \( \log 12 \), simply press the \( \log \) button followed by 12, and then press enter. The result is approximately 1.0792.
To find \( \log (1 / 4) \), enter the fraction as \( 0.25 \) and use the \( \log \) button. This should give you approximately -0.6021.

Using a calculator this way ensures accuracy and saves time, especially when solving complex logarithmic expressions. Always double-check your inputs to avoid mistakes.
Approximation
Approximation in mathematics means finding a value that is close enough to the exact answer, typically within an accepted range. When using calculators to find logarithmic values, the results are often rounded to a certain number of decimal places. In this exercise, the logarithms are approximated to four decimal places. For example, \( \log 12 \ \approx 1.0792 \ \) and \( \log (1 / 4) \ \approx -0.6021 \ \).

These approximations are sufficient for most practical purposes. However, if higher accuracy is needed, more decimal places can be used. Always ensure you understand the required precision for your calculations to achieve the best results.

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Most popular questions from this chapter

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{2 x}=4 $$

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