Problem 22 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The expression condenses to \(\log(x^2(x+1)^3)\).

Step 1: Apply the Power Rule

The Power Rule of logarithms states that for any logarithm of the form \(a \log b\), it can be rewritten as \(\log(b^a)\). First, apply this rule to transform \(2 \log(x)\) into \(\log(x^2)\). Similarly, transform \(3 \log(x+1)\) into \(\log((x+1)^3)\).

Step 2: Apply the Product Rule

The Product Rule of logarithms states that \(\log(a) + \log(b) = \log(ab)\). Now, apply this rule to combine \(\log(x^2) + \log((x+1)^3)\) into a single logarithm: \(\log(x^2(x+1)^3)\).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a helpful tool when dealing with logarithmic expressions that involve multiplication. The rule tells us that if you have a logarithm of a base raised to an exponent, that exponent can be brought in front as a multiplier. For example, a logarithm in the form of \( a \log b \) can be rewritten using the power rule as \( \log(b^a) \).
This means that instead of multiplying the logarithm by a constant, we can raise the argument of the logarithm to the power of that constant.

• This transformation simplifies expressions by changing multiplicative terms into exponential terms.
• For example, \( 2 \log(x) \) becomes \( \log(x^2) \).

Similarly, applying the power rule to \( 3 \log(x+1) \) results in \( \log((x+1)^3) \). It allows us to handle complex logarithmic expressions more smoothly, ultimately making them easier to work with in further simplifications.

Product Rule of Logarithms

The product rule of logarithms deals with the addition of two logarithms, allowing us to condense them into a single logarithmic expression. According to the product rule, when you add logarithms, you can multiply the numbers inside the log functions. In mathematical terms, \( \log(a) + \log(b) = \log(ab) \).
This property is particularly useful for consolidating expressions that are in the form of added logs.

• By using this rule, we transform addition between log expressions into multiplication inside a single log.
• For instance, combining \( \log(x^2) \) and \( \log((x+1)^3) \) using the product rule yields \( \log(x^2(x+1)^3) \).

This method simplifies complex expressions and is crucial for condensing multiple logarithms into a cleaner form.

Condensing Logarithmic Expressions

Condensing logarithmic expressions means bringing together multiple logarithms into a single, more manageable expression. This process often involves utilizing the properties of logarithms, such as the power rule and the product rule, to combine and simplify logarithmic terms.
Let's look at the expression \( 2 \log(x) + 3 \log(x+1) \). To simplify or condense it:

• You first apply the power rule to each term individually.
• Convert \( 2 \log(x) \) into \( \log(x^2) \) and \( 3 \log(x+1) \) into \( \log((x+1)^3) \).
• Then, employ the product rule to combine these into a single expression: \( \log(x^2(x+1)^3) \).

This process makes the expression simpler to handle and easier to integrate into further mathematical calculations or solutions. Condensing is particularly important for solving logarithmic equations and inequalities, where streamlined expressions lead to more effective problem-solving.