Problem 268 - Precalculus

Question

For the following exercises, condense each expression to a single logaritim using the properties of logaritims. $$ \log \left(2 x^{4}\right)+\log \left(3 x^{5}\right) $$

Step-by-Step Solution

Answer

The expression condenses to \( \log(6x^9) \).

Step 1: Use the Product Property of Logarithms

The product property of logarithms states that \( \log(a) + \log(b) = \log(ab) \). We need to apply this property to combine the two logarithmic expressions into one. Here, \( a = 2x^4 \) and \( b = 3x^5 \). Thus, the expression becomes \( \log((2x^4)(3x^5)) \).

Step 2: Multiply the Expressions Inside the Logarithm

Now, multiply the expressions inside the logarithm: \( (2x^4) \times (3x^5) \). Multiply the constants: \( 2 \times 3 = 6 \). For the same base \( x \), add the exponents: \( x^4 \times x^5 = x^{4+5} = x^9 \). Therefore, the expression inside the logarithm is \( 6x^9 \).

Step 3: Combine into a Single Logarithmic Expression

The expression \( \log((2x^4)(3x^5)) \) simplifies to \( \log(6x^9) \) based on the previous multiplication. Therefore, the condensed form of the given expression is \( \log(6x^9) \).

Key Concepts

Product Property of Logarithms

The product property of logarithms is a fundamental rule that helps simplify expressions with multiple logarithms. It states that when you add two logarithms with the same base, you can combine them into a single logarithm by multiplying their arguments. This property can be written as:

• \( \log(a) + \log(b) = \log(ab) \)

Think of it as adding two pieces together under one roof, making calculations simpler. For instance, in the given exercise where you have \( \log(2x^4) + \log(3x^5) \), you apply the product property to combine them into \( \log((2x^4)(3x^5)) \).
This results in a single, more manageable logarithmic expression.

By understanding and using the product property, you can streamline multiple logarithmic additions into one neat expression, setting the stage for further simplifications.

Exponent Rules in Logarithms

Exponent rules can also be crucial when dealing with logarithms, especially when working with expressions that include variable terms raised to a power. These rules allow for easy manipulation and simplification of expressions. A key exponent rule is:

• When multiplying like bases, you add the exponents: \( x^m \times x^n = x^{m+n} \).

This rule is essential in our exercise when multiplying the terms inside the logarithm \((2x^4) \times (3x^5)\). Apply the exponent rule to combine the powers of \(x\), resulting in \(x^{4+5} = x^9\).
Thus, the expression becomes \(6x^9\), by first multiplying the constant values and then using the exponent rule for the variable part.

Mastering these rules allows for efficient simplification, making complex logarithmic expressions easier to handle.

Condensing Logarithms

Condensing logarithms involves combining several logarithmic terms into one within an expression. This process uses different logarithmic properties to simplify complex logarithmic equations efficiently.

• Begin by identifying common bases and operators, such as the product or quotient.
• Use properties like the product property to merge expressions.

In our specific exercise, the task is to condense \( \log(2x^4) + \log(3x^5) \) into a single logarithm. By applying the product property, you merge these into \( \log(6x^9) \).
This involves multiplying the coefficients first, then adding the exponents for \(x\), as previously explained.

Condensing logarithms is a useful skill that simplifies expressions, assists in solving logarithmic equations, and reduces potential errors in computation.