Problem 71 - College Algebra

Question

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{5} 13 $$

Step-by-Step Solution

Answer

The value of \(\log_{5}13\) up to four decimal places is 1.5392

Step 1: Convert to a common or natural logarithm

To make the calculation easier, we can convert the base 5 logarithm into a common or natural logarithm using the change of base formula: \[ \log_{b}a = \frac{\log a}{\log b} \] Applying this formula, we get \[ \log_{5}13 = \frac{\log 13}{\log 5} \]

Step 2: Calculate with a calculator

Now, use a calculator to calculate the value of \(\frac{\log 13}{\log 5}\). Make sure the calculator is set to compute common (base 10) or natural (base e) logarithms. To get a result to four decimal places, round off your answer accordingly.

Key Concepts

Change of Base Formula

The change of base formula is a useful tool when dealing with logarithms of different bases. It allows you to convert any logarithm into a form that uses a more common base, typically 10 (common logarithms) or the natural base, e (natural logarithms). This conversion is particularly helpful when you are using a calculator, as most calculators are equipped to handle only these two types of logarithms.

• The formula is given by: \( \log_b a = \frac{\log_c a}{\log_c b} \).
• Here, \( b \) is your original base, \( a \) is the number you are taking the logarithm of, and \( c \) is the new base you want to change to (commonly 10 or e).

In practice, this means that you can compute any logarithm using familiar tools, either as a fraction of common or natural logarithms. This makes calculations more straightforward, without requiring special methods for every different logarithm base.

Common Logarithms

Common logarithms are logarithms with base 10, denoted as \( \log \, x \). They are extensively used in science and engineering due to their correlation with the decimal system. When no base is mentioned, it often implies base 10.

• For example, \( \log_{10} 100 = 2 \) because 10 squared equals 100.
• Common logs simplify the process of working with large numbers by converting them into manageable figures.

Most calculators feature a dedicated "log" button for common logarithms, allowing for quick calculations like \( \frac{\log 13}{\log 5} \). This is advantageous when working through exercises or solving real-world problems where precise base-10 logarithmic values are necessary.

Natural Logarithms

Natural logarithms, denoted \( \ln \, x \), utilize the constant base \( e \approx 2.718 \). They appear broadly in mathematics, especially in calculus and exponential growth problems.

• The primary distinction of the natural logarithm is its foundational relationship with the exponential function, where \( \ln(e) = 1 \).
• It reflects growth processes, such as population growth and radioactive decay, due to the base \( e \)'s unique mathematical properties.

Calculators typically offer an "ln" button for computing natural logarithms efficiently. This feature facilitates calculations involving phenomena modeled by exponential functions, providing a seamless approach for both academic and practical applications.