Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms.$$\log _{5} x$$.

Step-by-Step Solution

Answer

(a) Common logarithm form: \( \frac{\log x}{\log 5} \) (b) Natural logarithm form: \( \frac{\ln x}{\ln 5} \)

Step 1: Rewrite Using Change of Base Formula (Common Log)

Common logarithms have base 10. Rewrite the statement \( \log_{5} x \) in terms of base 10 using the change of base formula. This will yield \( \log_{5} x = \frac{\log_{10} x}{\log_{10} 5} \)

Step 2: Use the Property of Base 10 Logarithm

Remember that a base-10 logarithm can be written without the explicit base. So, \( \log_{10} x \) is written as \( \log x \). Thus, the final answer in terms of common logarithms is \( \frac{\log x}{\log 5} \) .

Step 3: Rewrite Using Change of Base Formula (Natural Log)

Natural logarithms have base \( e \) (the mathematical constant approximately equal to 2.71828). So, rewriting the original statement \( \log_{5} x \) in terms of natural logarithms, apply the change of base formula to get \( \log_{5} x = \frac{\ln x}{\ln 5} \) .

Key Concepts

Common Logarithms

When you encounter a logarithmic function such as \( \text{log}_{5}x \), understanding the concept of common logarithms is essential. A common logarithm is a logarithm with base 10 and is denoted simply as \( \text{log} \) without the base. In other words, when you see \( \text{log} x \), it is understood to be \( \text{log}_{10} x \).

This is important when using the change of base formula—a method allowing the calculation of logarithms with any base using a calculator which typically only has buttons for common (base 10) and natural (base e) logarithms. The formula is given by \( \text{log}_{b} a = \frac{\text{log} a}{\text{log} b} \), and for the given exercise, it helps to find the value of \( \text{log}_{5}x \) by expressing it as a ratio \( \frac{\text{log} x}{\text{log} 5} \).

Remember that common logarithms are widely used in various scientific fields like chemistry and engineering due to their historical computation convenience before the widespread use of electronic calculators.

Natural Logarithms

While common logarithms use base 10, natural logarithms use the special mathematical constant \( e \), approximately equal to 2.71828. In logarithmic terms, the natural logarithm of a number is the power to which \( e \) must be raised to yield that number. Its notation is \( \text{ln} \), distinguishing it from the common logarithm.

In the given exercise, you can also express \( \text{log}_{5}x \) by using natural logarithms. Applying the change of base formula again, you get \( \text{log}_{5} x = \frac{\text{ln} x}{\text{ln} 5} \). This form is useful in more advanced mathematics, particularly in calculus, where the natural logarithm plays a significant role due to its relationship with the rate of growth and areas under curves.

Logarithmic Properties

Understanding logarithmic properties can simplify the process of working with logarithms and can be a powerful tool for solving logarithmic equations. There are several key properties that are particularly useful:

• The product rule: \( \text{log}_{b}(xy) = \text{log}_{b}x + \text{log}_{b}y \)
• The quotient rule: \( \text{log}_{b}(\frac{x}{y}) = \text{log}_{b}x - \text{log}_{b}y \)
• The power rule: \( \text{log}_{b}(x^c) = c \text{log}_{b}x \)
• Change of base rule as discussed earlier: \( \text{log}_{b}a = \frac{\text{log}_{c}a}{\text{log}_{c}b} \) for any positive base \( c \), not equal to 1.

These properties can significantly facilitate the simplification and manipulation of logarithmic expressions and are essential for any student looking to master logarithms for higher level math.