Problem 89 - A Graphical Approach to College Algebra

Question

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{5} 10$$

Step-by-Step Solution

Answer

\(\log_5 10 \approx 1.43067\).

Step 1: Understanding the Change-of-Base Formula

The change-of-base formula allows us to evaluate logarithms using a different base, usually 10 or e, which are easier to compute with calculators. The formula is: \[\log_b a = \frac{\log_c a}{\log_c b}\]where \(c\) is the new base. Common choices for \(c\) are 10 (common logarithm) or \(e\) (natural logarithm).

Step 2: Apply the Change-of-Base Formula

We need to find \(\log_5 10\). Using the change-of-base formula with base 10, we write:\[\log_5 10 = \frac{\log_{10} 10}{\log_{10} 5}\]

Step 3: Calculate Each Logarithm Using a Calculator

Using a calculator, we find: \(\log_{10} 10 = 1\) because 10 raised to the power of 1 is 10, and \(\log_{10} 5 \approx 0.69897\).

Step 4: Compute the Approximation

Now, substitute the values into the change-of-base formula:\[\log_5 10 = \frac{1}{0.69897} \approx 1.43067\] So, \(\log_5 10 \approx 1.43067\).

Key Concepts

Logarithms

Logarithms are a mathematical operation that help us deal with exponential relationships. They are essentially the inverse of exponentiation. This means if you know a result in an exponential equation and you want to find one of the exponents, you would use logarithms. For example, in the equation \( 10^x = 100 \), we can find \( x \) by using logarithms.We have different bases for logarithms:

• Logarithm base 10: This is known as the common logarithm. Often written as \( \log \).
• Logarithm base \( e \): This is known as the natural logarithm. Represented as \( \ln \).
• Logarithm base 2: Used in computing, known as binary logarithm.

By switching bases using a formula known as the change-of-base formula, we can calculate logarithms more easily, particularly when using calculators that have built-in functions for specific bases.

Common Logarithm

The common logarithm is simply a logarithm with a base of 10. It's widely used in science and engineering because it simplifies expressions with powers of 10. When you see \( \log \) without a specified base, it's usually assumed to be base 10.One main property of the common logarithm is:

• \( \log_{10} 10 = 1 \): because 10 to the power of 1 is 10.

You will often use common logarithms in your calculations involving the change-of-base formula. This is because most scientific calculators have a dedicated \( \log \) button that automatically uses base 10. As a result, using common logarithms makes computations straightforward when evaluating logarithms with different bases.

Natural Logarithm

Natural logarithms use the base \( e \), where \( e \approx 2.718 \). This base, \( e \), arises naturally in the context of continuous growth or decay, making natural logarithms very important in calculus and complex mathematics.Whenever you see \( \ln \), it denotes a logarithm with base \( e \). Some key characteristics include:

• The natural logarithm of \( e \), \( \ln e = 1 \), because \( e^1 = e \).
• They are useful in solving differential equations and situations involving compound interest.

Natural logarithms provide a framework for problems involving rates of exponential growth, which appear in fields ranging from natural sciences to economics. Understanding the natural logarithm allows you to connect exponential functions to their inverses, simplifying complex calculations.