Problem 31 - Precalculus

Question

Use the appropriate change of base formula to convert the given expression to an expression with the indicated base. $$ \log _{3}(x+2) \text { to base } 10 $$

Step-by-Step Solution

Answer

\(\log_{3}(x+2) = \frac{\log_{10}(x+2)}{\log_{10}(3)}\) using base 10.

Step 1: Understand the Change of Base Formula

The change of base formula allows you to rewrite a logarithmic expression in terms of logs with a different base. The formula is given by: \[\log_b a = \frac{\log_c a}{\log_c b},\]where \(b\) is the original base, \(c\) is the new base, and \(a\) is the argument of the logarithm.

Step 2: Identify Components of the Expression

We need to convert \(\log_{3}(x+2)\) to base 10 using the change of base formula. Here, \(b = 3\) is the original base, \(a = x + 2\) is the argument of the logarithm, and \(c = 10\) is the new base.

Step 3: Apply the Change of Base Formula

Substitute the identified components into the change of base formula:\[\log_{3}(x+2) = \frac{\log_{10}(x+2)}{\log_{10}(3)}.\]This expression uses logarithms with base 10 to represent \(\log_{3}(x+2)\).

Key Concepts

Logarithmic Expression

A logarithmic expression is a mathematical way of expressing the power to which a base must be raised to obtain a given number. It is an important concept in mathematics, especially in areas related to exponential growth and decay. An expression like \(log_{b}(a)\) can be thought of as asking: "To what power does the base \(b\) need to be raised to yield \(a\)?" For example, if \(b = 3\) and \(a = 27\), \(log_{3}(27)\) would be 3, since 3 to the power of 3 is 27.
When you are working with logarithmic expressions, it helps to think of them as the inverse operation to exponentiation. Just as addition is the opposite of subtraction, and multiplication is the opposite of division, logarithms undo the process of exponentiation.
By understanding logarithms, you can solve complex equations involving exponential relationships, making them a powerful tool in both pure and applied mathematics.

Base Conversion

Base conversion in mathematics allows us to change the base of a logarithmic expression to make calculations easier or to better suit a given problem. The Change of Base Formula is a simple yet powerful tool. It shows us how to convert a logarithm of any base \(b\) to another base \(c\). The formula is:\[\log_b a = \frac{\log_c a}{\log_c b}\]You just take the logarithm of the argument with the new base and divide it by the logarithm of the original base with the new base. This formula is particularly handy when you want to use a calculator that might only have keys for specific bases, like base 10 or base \(e\) (the natural logarithm).
For example, to convert \(\log_{3}(x+2)\) to a base 10 expression, you would use the formula to write it as:\[\frac{\log_{10}(x+2)}{\log_{10}(3)}\]By doing this conversion, you're essentially expressing the original logarithm in terms of something more manageable, allowing for easier computation or deeper analysis.

Precalculus Concepts

In precalculus, students often encounter various mathematical expressions and learn how to manipulate them. One crucial concept is logarithms and the change of base. These concepts not only serve as a bridge to calculus but also introduce a necessary set of tools for solving more complex mathematical problems.
This transformation from one base to another might seem like a small mathematical trick, but it's a powerful technique that highlights the interconnectivity of mathematical principles. Precalculus covers the fundamentals that allow you to perform these operations seamlessly. It's important because it builds the foundation you need to handle equations with exponential and logarithmic components confidently.
Having a good grasp on how to work with different bases empowers you to understand different mathematical scenarios better, providing flexibility in problem-solving. Whether preparing for calculus or simply looking to improve your mathematical skills, mastering these precalculus concepts is key to unlocking higher-level math and its applications in various fields.