Problem 7 - Precalculus with Limits

Question

Rewriting a Logarithm In Exercises \(7-10\) , rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{5} 16$$

Step-by-Step Solution

Answer

(a) \(\log_{5} 16 = \frac{\log_{10} 16}{\log_{10} 5}\) (b) \(\log_{5} 16 = \frac{\ln 16}{\ln 5}\)

Step 1: Use of change-of-base formula to 10

Apply the change-of-base formula to convert the base to 10. So, \(\log_{5} 16 = \frac{\log_{10} 16}{\log_{10} 5}\).

Step 2: Use of change-of-base formula to e

Apply the change-of-base formula to convert the base to \(e\). So, \(\log_{5} 16 = \frac{\ln 16}{\ln 5}\).

Key Concepts

Logarithms

A logarithm is a fundamental concept in mathematics that helps us understand how many times a certain number, called the "base," must be multiplied by itself to reach another number. Essentially, if you have a logarithm expressed as \(log_b a\), it tells us the power \(n\) that the base \(b\) must be raised to produce the number \(a\). For example, in \(log_2 8 = 3\), the base 2 is raised to the power of 3 to get 8.
Logarithms simplify calculations as they transform multiplication into addition. As a result, they're extremely useful in various scientific fields and are integral to equations in algebra, calculus, and beyond.
Moreover, logarithms help us deal with exponential growth or decay, such as in finance when calculating compound interest, in science when dealing with pH levels in chemistry, or even measuring earthquake intensities.

Common Logarithms

Common logarithms are specific types of logarithms that use 10 as their base, denoted as \(log_{10} x\) or sometimes simply \(log x\). These are widespread in real-life scenarios, primarily because our number system is based on 10.

• Simplification: By using base 10, complex multiplications become simpler adds.
• Convenience: Scientific calculators and logarithmic tables often default to base 10, hence it's convenient for quick calculations.
• Examples: The sound intensity measured in decibels uses the common logarithm.

A practical application of converting logarithms with different bases to common logarithms uses the change-of-base formula: \(log_b a = \frac{\log_{10} a}{\log_{10} b}\). This formula allows the rewrite of logarithms, like the original exercise problem \(\log_5 16\) as common logarithms: \(\frac{\log_{10} 16}{\log_{10} 5}\). This approach makes it easier since calculators usually support base 10 directly.

Natural Logarithms

Natural logarithms use the base \(e\), a mathematical constant approximately equal to 2.71828. They are denoted \(ln x\) where \(e\) is the base, which is irrational and transcendent, meaning it can't be expressed as a simple fraction.

• Natural Growth: Natural logarithms are particularly valuable in situations involving continuous growth or decay, like radioactive decay or population growth models.
• Integration with Calculus: As \(e\) is intricately linked with calculus, natural logarithms frequently appear in calculus-based solutions.
• Conversion and Use: Using the change-of-base formula, we can easily convert any logarithm to a natural logarithm. For instance, \(\log_5 16\) can be expressed using natural logarithms as \(\frac{\ln 16}{\ln 5}\).

Calculators also support natural logarithm calculations, making these forms both practical and essential in advanced mathematical topics and real-world applications where growth follows natural patterns.