Problem 76 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

Understanding Common Logarithms

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