Problem 76 - College Algebra Essentials

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

This can be calculated as either \(\frac{\log_{10} 19}{\log_{10} 0.3}\) if using common logarithms or as \(\frac{\ln{19}}{\ln{0.3}}\) if using natural logarithms. Remember to round to four decimal places when using your calculator.

Step 1: Understand the change of base formula

The change of base formula is defined as \(\log_b a = \frac{\log_k a}{\log_k b}\), where a, b, and k are positive real numbers, b ≠ 1, and k ≠ 1. Here, b is the base of the logarithm, a is the number for which the logarithm is being taken, and k is any other chosen base that is convenient for calculation.

Step 2: Insert the given values into the formula

By substituting the given values into the change of base formula, we get \(\log_{0.3} 19 = \frac{\log_k 19}{\log_k 0.3}\), either where k = 10 for common logarithms or k = e for natural logarithms.

Step 3: Calculate using a calculator

If using common logarithms, the calculation is \(\frac{\log_{10} 19}{\log_{10} 0.3}\). If using natural logarithms, the calculation is \(\frac{\ln{19}}{\ln{0.3}}\). These calculations can be carefully done using a scientific calculator, remembering to round to four decimal places.

Key Concepts

Change of Base Formula

The change of base formula is a handy tool for evaluating logarithms, particularly when the base is not 10 or \(e\). Especially in cases where your calculator is limited to computing only common logarithms (base 10) or natural logarithms (base \(e\)), this formula shines.

• To use the change of base formula, identify the base \(b\) of the logarithm you want to compute.
• Next, choose a more calculation-friendly base such as 10 or \(e\), noted as \(k\).
• Input your values into the formula: \(\log_b a = \frac{\log_k a}{\log_k b}\).

This essentially means your chosen base \(k\) replaces the original base, breaking it down into manageable parts that a standard calculator can handle.Remember, \( k \) can be 10 for common logarithms or \( e \) for natural logarithms, offering flexible options when working with any bases.

Common Logarithms

Common logarithms are logarithms with a base of 10, commonly denoted as \( \log \). You utilize common logarithms in a range of applications, particularly when dealing with scientific calculations, as they frequently relate to scale measurements like the Richter scale for earthquakes or decibels for sound intensity.

• Notation for common logarithms is \( \log_{10} \) or often just \( \log \).
• Calculators are typically equipped to handle common logarithms directly via a dedicated log button.
• The change of base formula applies just as easily by setting \( k = 10 \).

For example, to calculate \( \log_{0.3} 19 \) using common logarithms, you would use \( \frac{\log_{10} 19}{\log_{10} 0.3} \) and enter it into your calculator. Calculating this will give you an approximate value rounded to four decimal places.

Natural Logarithms

Natural logarithms use a base of \(e\), where \(e\) is Euler's number and is approximately equal to 2.71828. They are widely used in higher-level mathematics and natural sciences as \(e\) is the foundation for continuous growth or decay.

• Notation for natural logarithms is \( \ln \), signifying \( \log_e \).
• Using natural logarithms is applicable and sometimes preferred in calculus and exponential growth scenarios.
• Most calculators have an \( \ln \) button for easy computation.

To compute \( \log_{0.3} 19 \) using natural logarithms, you simply apply the change of base formula as \( \frac{\ln 19}{\ln 0.3} \). This method yields the same exactness as using common logarithms but often provides more contextual appropriateness in certain mathematical fields.