Problem 10 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms.$$\log _{a} \frac{4}{5}$$.

Step-by-Step Solution

Answer

\(\log_a\frac{4}{5} = \frac{\log\frac{4}{5}}{\log a} \text{ (common log version)}, \frac{\ln\frac{4}{5}}{\ln a} \text{ (natural log version)}\)

Step 1: Apply the Change of Base Formula using base 10 logs

The change of base formula is \[ \log_b a = \frac{\log a}{\log b}\]. For part (a), apply this formula using base 10 logs, denoted \(\log\), to get \(\log_a\frac{4}{5} = \frac{\log\frac{4}{5}}{\log a}\).

Step 2: Apply the Change of Base Formula using natural logs

For part (b), apply the change of base formula using base 'e' logs, denoted ln, to get \(\log_a\frac{4}{5} = \frac{\ln\frac{4}{5}}{\ln a}\).

Key Concepts

Change of Base Formula

The change of base formula is an invaluable tool when working with logarithms of different bases. In many calculators and computer programs, you often only have the functionality for logarithms of base 10 or base 'e', commonly referred to as common and natural logarithms, respectively. The change of base formula allows us to express any logarithm in terms of these more standard bases.

Here's the general idea: for any logarithm base 'b', you can express \[ \log_b x = \frac{\log_k x}{\log_k b} \] where 'k' is your new base of choice, such as 10 or 'e'. By applying this formula, you're able to calculate logarithms for any base using your calculator’s built-in functions.

This formula essentially breaks down the original logarithm into a ratio of logs with a common base, which can be more easily computed. This approach is particularly useful in scenarios where you need to compare logarithms with different bases or use technology that doesn't support arbitrary base computations.

Common Logarithms

Common logarithms are logarithms that use 10 as their base. They are often simply written as \(\log\), without specifying the base formally. As one of the most straightforward and frequently used logarithmic calculations, common logarithms are central in many scientific fields, especially where measurements span large orders of magnitude.

• Base: 10
• Standard notation: \(\log x\)
• Useful for power of ten scales, such as the pH scale, decibels, and Richter scale.

When using the change of base formula to go with common logarithms, you're utilizing the base 10 logarithm to express logs with various bases. This can easily be done with most calculators and simplifies complex logarithmic problems, making them approachable with the tools at hand.

Natural Logarithms

Natural logarithms focus on a mathematic constant, 'e', approximately 2.718. Denoted as ln, they hold significant importance in higher mathematics and natural sciences due to their unique properties in calculus, particularly when discussing growth processes and complex models.

• Base: \(e\)
• Standard notation: \(\ln x\)
• Integral in fields like biology and financial modeling due to their relationship with exponential growth and decay.

The use of natural logarithms through the change of base formula provides flexibility in computations and derivations, especially when you're dealing with problems involving exponential growth, such as population dynamics or radioactive decay. By converting a logarithmic equation into one that uses natural logs, you can harness the computational power of this mathematical constant to reveal new insights into problems that may be overly complex otherwise.