Problem 22 - College Algebra

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

\( \log (x^2 (x+1)^3) \)

Step 1: Apply Power Rule

Use the power rule of logarithms, which states that \( a \log b = \log b^a \). Apply this to each term: \( 2 \log (x) = \log (x^2) \) and \( 3 \log (x+1) = \log ((x+1)^3) \).

Step 2: Apply Product Rule

Apply the product rule of logarithms, \( \log a + \log b = \log (a \cdot b) \). Combine the two logarithms: \( \log (x^2) + \log ((x+1)^3) = \log (x^2 \cdot (x+1)^3) \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool when you need to simplify expressions in which the logarithm is multiplied by a coefficient. It is based on the principle that multiplying a logarithm by a constant is the same as raising the argument of the logarithm to a power. In mathematical terms, the power rule says: \( a \log b = \log (b^a) \). Here's how it works: When you see an expression like \( 2 \log (x) \), apply the power rule by moving the coefficient (in this case, 2) to the exponent of \( x \). So, \( 2 \log (x) \) becomes \( \log (x^2) \). It's that simple! This way, you're turning the multiplication of the logarithm into an exponent, which is often easier to work with.

• Move the coefficient to the exponent: \( a \log b \rightarrow \log (b^a) \).
• Example: \( 3 \log (x+1) \rightarrow \log((x+1)^3) \).

Product Rule of Logarithms

The product rule of logarithms is useful for combining two logarithms into one. It helps when you have a sum of logarithms, each with a different argument, and you want to express them as a single logarithm. The rule states:\( \log a + \log b = \log (a \cdot b) \). This means that if you are adding two logarithms, you can multiply the arguments, or the numbers inside the logarithms, and use one single logarithm for the result. For example, if you have \( \log (x^2) \) and \( \log ((x+1)^3) \), you can combine them into one logarithm: \( \log (x^2 \cdot (x+1)^3) \).Understanding this rule helps in simplifying expressions or solving equations involving logarithms, making complex tasks easier to handle. Always remember:

• Addition to multiplication: Turning addition of logarithms into multiplication of arguments.
• Example: \( \log a + \log b = \log (a \cdot b) \).

Condensing Logarithms

Condensing logarithms means simplifying an expression that involves multiple logarithms into a single, more compact form. This is done by using the properties of logarithms, such as the power rule and the product rule, to combine terms. The process involves stepping through each property systematically. First, apply the power rule to bring any coefficients in front of the logarithms into the exponents. For example, \( 2 \log (x) \) changes to \( \log (x^2) \). Second, use the product rule to combine the terms into a single logarithm as in \( \log (x^2) + \log ((x+1)^3) = \log (x^2 \cdot (x+1)^3) \).Ultimately, condensing logarithms helps in creating simpler expressions that are often easier to work with in solving further mathematical problems or when performing more operations. When condensing:

• Use the power rule for coefficients: \( a \log b \rightarrow \log (b^a) \).
• Use the product rule for addition: \( \log a + \log b = \log (a \cdot b) \).
• Resulting in: A single, combined logarithm that represents the original expression.