Problem 270 - Precalculus

Question

For the following exercises, condense each expression to a single logaritim using the properties of logaritims. $$ 2 \log (x)+3 \log (x+1) $$

Step-by-Step Solution

Answer

The condensed expression is \( \log (x^2 (x+1)^3) \).

Step 1: Apply Power Rule

The power rule of logarithms states that \( a \log b = \log (b^a) \). We can apply this to our expression: \( 2 \log (x) = \log (x^2) \) and \( 3 \log (x+1) = \log ((x+1)^3) \). So, the expression becomes: \( \log (x^2) + \log ((x+1)^3) \).

Step 2: Apply Product Rule

The product rule of logarithms states that \( \log a + \log b = \log (ab) \).Apply this to the transformed expression: \( \log (x^2) + \log ((x+1)^3) = \log (x^2 (x+1)^3) \).

Step 3: Condense the Expression

We have successfully used the properties of logarithms to condense the expression:\( \log (x^2 (x+1)^3) \).

Key Concepts

Understanding the Power Rule of Logarithms

The power rule of logarithms is a key concept that can simplify complex logarithmic expressions. This rule states that multiplying a logarithm by a constant is equivalent to raising the number inside the log to the power of that constant.
For instance, if you have an expression like \( a \log b \), you can rewrite it as \( \log (b^a) \).

• Think of it as transferring the constant \( a \) from a coefficient outside the log to the exponent inside the log.
• This transformation helps reduce and simplify expressions, especially when condensing multiple logarithms.

In the exercise provided, the power rule is applied to transform \( 2 \log (x) \) and \( 3 \log (x+1) \) into \( \log (x^2) \) and \( \log ((x+1)^3) \) respectively.
This step is crucial as it sets the stage for further simplification using other logarithmic properties.

Mastering the Product Rule of Logarithms

After using the power rule, we often proceed with the product rule of logarithms to combine terms. The product rule indicates that adding two logarithms with the same base equates to taking the logarithm of the product of their arguments.
In mathematical terms, \( \log a + \log b = \log (ab) \).

• This rule allows you to combine multiple log terms into a single, more manageable expression.
• It becomes particularly useful when you want to condense logarithmic expressions by combing them based on their multiplication properties.

Applying the product rule in our example, we started with \( \log (x^2) + \log ((x+1)^3) \).
By using the product rule, these were combined into a single expression: \( \log (x^2 (x+1)^3) \).
This simplifies our task of working with the logarithmic expressions significantly, making it easier to solve or analyze further.

Condensing Logarithmic Expressions Efficiently

Condensing logarithmic expressions involves using logarithm properties to simplify the expression into a single log. This is particularly useful for complex expressions and requires the effective application of both the power and product rules.
The aim is to turn an expression like \( 2 \log (x) + 3 \log (x+1) \) into a single log statement: \( \log (x^2 (x+1)^3) \).

• Start by applying the power rule to deal with coefficients and exponents.
• Then, use the product rule to merge addition between log terms into one unified product expression within a log.

The process of condensing is not just about simplifying; it's also about making the equations easier to manage and solve.
By following these steps systematically, any complex logarithmic expression can be efficiently restructured into a more straightforward form.
Mastering condensing techniques involves practice and a solid understanding of how each logarithm property interacts with the others.