Problem 71 - Algebra and Trigonometry

Question

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{5} 13 $$

Step-by-Step Solution

Answer

The approximate value of \( \log _{5} 13 \) to four decimal places is 1.6395.

Step 1: Understand the Problem

We need to find the value of \( \log _{5} 13 \) to four decimal places. Since calculators usually only calculate to base 10 (common logarithms) or base e (natural logarithms), we need to convert the base.

Step 2: Convert the Base using Change of Base Formula

The change of base formula is \( \log_{b} a = \frac{\log a}{\log b} \). Using this formula, \( \log _{5} 13 \) can be rewritten as \( \frac{\log 13}{\log 5} \) or \( \frac{ln 13}{ln 5} \).

Step 3: Evaluate using a Calculator

After converting to common or natural logarithms, use a calculator to find the value. Round the result to four decimal places.

Key Concepts

Change of Base Formula

When it comes to calculating logarithms with bases other than 10 or 'e', the change of base formula becomes an essential tool. It allows us to convert complex logarithms into ones that can be easily evaluated using a standard scientific calculator. The formula is given by:
\[ \log_{b} a = \frac{\log a}{\log b} \]or
\[ \log_{b} a = \frac{\ln a}{\ln b} \]
Using the change of base formula, any logarithm can be expressed in terms of common (base 10) or natural (base 'e') logarithms which are readily accessible functions on most calculators. For example, to calculate \( \log_{5} 13 \), you can rewrite it as \( \frac{\log 13}{\log 5} \) or \( \frac{\ln 13}{\ln 5} \), which simplifies the computation to basic arithmetic with the results given by your calculator. This approach is practical and widely used, especially when dealing with logarithms of arbitrary bases.

Common Logarithms

Common logarithms are the logarithms with base 10, and they are usually denoted as \( \log \) without a base. In simpler terms, the common logarithm of a number answers the question: 'To what power must 10 be raised, to produce this number?'
For instance, \( \log 100 \) is 2 because \( 10^2 = 100 \). Most calculators have a dedicated button for common logarithms, labeled as 'log'. Since base 10 is so integral to our number system (decimals, scientific notation, etc.), common logarithms are frequently encountered in various fields of science and engineering. Understanding how to work with common logarithms, including using them to solve problems through the change of base formula, is a fundamental skill in mathematics.

Natural Logarithms

Natural logarithms are another crucial type of logarithm in mathematics, especially calculus. They have a base of 'e', where 'e' is an irrational and transcendent number approximately equal to 2.71828. The notation for natural logarithms is \( \ln \), so when you see \( \ln x \), it's asking for the power to which 'e' must be raised to get the number 'x'.
For example, since 'e' raised to the power of 1 equals 'e', \( \ln e = 1 \). Natural logarithms are omnipresent in continuous growth models, complex financial calculations, and many areas of science. Many calculators feature the 'ln' button, allowing for direct computation of natural logarithms. They are inherently related to exponential functions and are invaluable in solving problems involving continuous growth or decay.

Evaluating Logarithms

Evaluating logarithms may seem daunting at first, but by using tools like the change of base formula and understanding the properties of common and natural logarithms, it becomes a manageable task. To evaluate a logarithm, you need to identify its base and match it to your calculation method: base 10, base 'e', or another base. If the base is not 10 or 'e', use the change of base formula as shown in our exercise above for \( \log_{5} 13 \). With the rewritten expression, you can then enter the values into the calculator.
Remember to round the result to the appropriate number of decimal places as required. Using the change of base formula does not alter the value of the logarithm; it simply translates it into a form that we can readily evaluate. Breaking down the process and understanding each step ensures that you can tackle any logarithmic expression confidently.