Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms.$$\log _{a} \frac{3}{10}$$.

Step-by-Step Solution

Answer

The logarithm \(\log _{a} \frac{3}{10}\) can be expressed as a ratio of common logarithms as \(\log _{a} \frac{3}{10} = \frac{\log \frac{3}{10}}{\log a}\) and as a ratio of natural logarithms as \(\log _{a} \frac{3}{10} = \frac{\ln \frac{3}{10}}{\ln a}\).

Step 1: Express as a Ratio of Common Logarithms

Applying the Change of Base Formula to the logarithm \(\log _{a} \frac{3}{10}\) with a base of 10 (common logarithm), this can be written as: \(\log _{a} \frac{3}{10} = \frac{\log \frac{3}{10}}{\log a}\).

Step 2: Express as a Ratio of Natural Logarithms

Similarly, applying the Change of Base Formula to the logarithm \(\log _{a} \frac{3}{10}\) with a base of e (natural logarithm), results in: \(\log _{a} \frac{3}{10} = \frac{\ln \frac{3}{10}}{\ln a}\).

Key Concepts

Common Logarithms

Common logarithms are the type of logarithms that have 10 as their base. They are denoted as \( \log x \), without the base written explicitly, because base 10 is implied. These logarithms are widely used because of their practicality in scientific calculations and because our number system is based on powers of 10. When compared to other bases, the properties of common logarithms, such as \( \log(mn) = \log m + \log n \) or \( \log(m/n) = \log m - \log n \), make them particularly helpful in simplifying mathematical expressions involving multiplication and division.

In the context of the exercise, to rewrite the logarithm \( \log _{a} \frac{3}{10} \) as a ratio of common logarithms, you can use the change of base formula. Thus, \( \log _{a} \frac{3}{10} = \frac{\log \frac{3}{10}}{\log a} \). This formula allows us to convert a logarithm with any base to a logarithm with base 10, facilitating easier computation, especially using calculators that have a common logarithm function.

Natural Logarithms

Natural logarithms are those that use the mathematical constant \( e \) (approximately equal to 2.71828) as their base and are represented as \( \ln x \). The constant \( e \) is unique in the way that it naturally arises from and is deeply connected to various areas of mathematics, including calculus, complex analysis, and differential equations. Natural logarithms are particularly useful when dealing with exponential growth or decay, such as in compound interest or radioactive decay.

When we apply the change of base formula to the calculation in the same exercise, changing from base \( a \) to the natural logarithm, we have \( \log _{a} \frac{3}{10} = \frac{\ln \frac{3}{10}}{\ln a} \). This transformation allows us to leverage the properties of natural logarithms, which can simplify the integration, differentiation, and solution of equations encountered in higher-level mathematics.

Logarithmic Expressions

Logarithmic expressions involve the logarithm function, which is the inverse of exponentiation. Understanding how to manipulate these expressions is key to solving various algebraic and calculus problems. A logarithmic expression contains the logarithm of a number or variable, which essentially asks, 'To what exponent must the base be raised to produce this number or variable?'.

Performing operations such as expanding, condensing, or changing the base of logarithmic expressions can often simplify them and make other operations more straightforward. For instance, in the exercise \( \log _{a} \frac{3}{10} \) can be viewed as the logarithmic expression involving base \( a \). Using the change of base formula is a method to rewrite this expression in terms of common or natural logarithms, making it easier to evaluate and understand. It's like translating a sentence into a more familiar language to comprehend its meaning more clearly.