Problem 33 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{10} x-\log _{10}(x+1)+\log _{10}\left(x^{2}-2\right) $$

Step-by-Step Solution

Answer

\( \log_{10}\left(\frac{x(x^2 - 2)}{x+1}\right) \)

Step 1: Understanding Logarithm Properties

In order to combine the given expression into a single logarithm, we will use the properties of logarithms. The most essential properties are:1. The product rule: \( \log_b(M) + \log_b(N) = \log_b(MN) \)2. The quotient rule: \( \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \)3. The power rule: \( k \log_b(M) = \log_b(M^k) \)

Step 2: Applying the Quotient Rule

We start by applying the quotient rule to the first two terms of the expression: \[ \log_{10} x - \log_{10}(x+1) = \log_{10}\left(\frac{x}{x+1}\right) \]

Step 3: Applying the Product Rule

Now, combine the result from Step 2 with the third term of the original expression using the product rule:\[ \log_{10}\left(\frac{x}{x+1}\right) + \log_{10}(x^2 - 2) = \log_{10}\left(\frac{x}{x+1} \cdot (x^2 - 2)\right) \]

Step 4: Writing as a Single Logarithm

The expression is now written as a single logarithm:\[ \log_{10}\left(\frac{x(x^2 - 2)}{x+1}\right) \].

Key Concepts

Product Rule

The product rule is an important and straightforward property of logarithms. It states that the sum of logarithms of two numbers is equal to the logarithm of the product of these numbers. In mathematical terms, this property is represented as: \( \log_b(M) + \log_b(N) = \log_b(MN) \).
This rule allows us to simplify expressions that involve the addition of logarithms. By recognizing which terms can be multiplied together, we can collapse them into a single logarithmic expression.
The product rule is extremely helpful when simplifying complex logarithmic equations, especially in exercises where we are asked to combine several logarithms into one. It simplifies calculations and is widely used in various mathematical contexts, such as algebra and calculus.

Quotient Rule

The quotient rule states that the difference of two logarithms with the same base can be rewritten as the logarithm of a quotient. Formally, the rule is expressed as \( \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \).
This property is specifically used when you have a subtraction between two logarithmic expressions. By applying the quotient rule, you can transform this subtraction into a division operation inside a single logarithm.
In problems like the original exercise, the quotient rule is the first step to reduce multiple logs into fewer terms, moving us closer to expressing the equation as a single logarithm. It's fundamental in simplifying expressions and solving logarithmic equations efficiently.

Power Rule

The power rule is another useful property of logarithms, which involves the manipulation of exponents. It can be expressed as: \( k \log_b(M) = \log_b(M^k) \).
This rule allows you to "move" a constant multiplier of a logarithm into an exponent. By doing so, you can adjust the function in a way that makes it possible to further simplify or manipulate it.
Even though the power rule wasn't directly applied in the original exercise provided, understanding it would be valuable for similar exercises, particularly those involving exponents within logarithmic terms. This rule helps streamline expressions into simpler forms, making the task of solving or integrating them more manageable.