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Chapter 4: Problem 76

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{0.3} 19 $$

Short Answer

Expert verified
Evaluating the expression and rounding to four decimal places, we find that \(\log _{0.3} 19 \) equals approximately -1.7859 using common logarithm or -1.7859 using natural logarithm. Since both common and natural logarithms provide the same decimal result, both answers are valid.

Step by step solution

01

Understanding the Change of Base Formula

The change of base formula is a mathematical formula used to change the base of a logarithm. The formula follows this structure: \(\log_b a = \frac{\log a}{\log b}\). In this case, b is the current base and a is the number. Here, 'log' represents the common logarithm (base 10) or natural logarithm (base e), which are typically what a calculator is set to handle.
02

Applying the Change of Base Formula

Let's replace a with 19 and b with 0.3 in the formula from Step 1 to change the base to 10 or e. This will translate the expression from \(\log _{0.3} 19 \) to \(\frac{\log 19}{ \log 0.3}\) if you're using the common logarithm or to \(\frac{\ln 19}{ \ln 0.3}\) if you're using the natural logarithm.
03

Evaluating the Logarithm

Now we will calculate these values using a calculator. Remember to round off the answer to four decimal places as instructed in the problem.

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Key Concepts

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

Understanding Common Logarithms
A common logarithm is the logarithm with base 10. It is denoted as \( \log x \) and literally asks the question, \

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Most popular questions from this chapter

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