Problem 63 - Elementary and Intermediate Algebra

Question

Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\log x^{3} y^{2}\)

Step-by-Step Solution

Answer

\(\log x^{3} y^{2} = 3\log(x) + 2\log(y)\) as a sum of logs.

Step 1: Understand the Power Rule for Logarithms

The power rule states that for any positive number \(a\) and exponent \(n\), \(\log(a^n) = n \cdot \log(a)\). This means if you have an exponent inside the logarithm, you can bring it to the front as a coefficient.

Step 2: Apply the Power Rule

Apply the power rule to each part of the expression \(\log x^{3} y^{2}\). Start by expressing it as \(\log(x^3) + \log(y^2)\) using the property that \(\log(ab) = \log(a) + \log(b)\). This gives two separate logarithmic expressions.

Step 3: Expand Each Logarithm

Based on the power rule, expand \(\log(x^3)\) to \(3\log(x)\) and \(\log(y^2)\) to \(2\log(y)\). The product of numbers inside a logarithm turns into the sum of two logs by bringing down the exponents of each term.

Step 4: Write the Final Expression

Combine the expanded logarithmic expressions from the previous step into a single expression: \(3\log(x) + 2\log(y)\). This is the expression of the original logarithm as the sum of logarithms.

Key Concepts

Power Rule for Logarithms

The power rule for logarithms is a fundamental property that helps us manipulate logarithmic expressions easily. If you have a logarithm where the argument is raised to an exponent, the power rule allows you to move that exponent in front of the logarithm as a coefficient. This can greatly simplify calculations. For example, the power rule states:

• \(\log(a^n) = n \cdot \log(a)\)

This means if you have \(\log(x^3)\), you can rewrite it as \(3 \cdot \log(x)\). This is especially useful when tackling complex expressions, as it lets you simplify them step-by-step. Keep this rule in your toolkit for all your logarithmic explorations. By understanding this rule, you can transform many logarithmic problems into simpler, more manageable forms.

Logarithmic Expressions

Logarithmic expressions involve logarithms, which are the inverse operations of exponentiation. Just like how multiplication is the opposite of division, logarithms help us undo exponentiation. This is why understanding logarithmic expressions is key to numerous mathematical calculations.When working with logarithmic expressions, we often come across different properties like the product, quotient, and power rules. An expression such as \(\log(x^3 y^2)\) can be daunting at first. But by applying these properties systematically, you can break down the expressions into simpler parts, like \(\log(x^3)+\log(y^2)\). This is done using another essential property:

• \(\log(ab) = \log(a) + \log(b)\)

With the power of these properties, you can express complex logarithmic terms as sums and differences, simplifying their complexity considerably.

Simplifying Logarithms

Simplifying logarithms can turn a potentially complicated expression into something much easier to handle. When you encounter an expression like \(\log(x^3 y^2)\), your goal is to express it as a combination of simpler logarithmic terms.First, recognize the structure: it is a product in exponential form. Use the properties of logarithms to expand it. Start by splitting the logarithm using the product rule, \(\log(ab) = \log(a) + \log(b)\), transforming the expression to \(\log(x^3) + \log(y^2)\). Next, apply the power rule we discussed earlier:

• \(\log(x^3) = 3\log(x)\)
• \(\log(y^2) = 2\log(y)\)

Once these transformations are completed, you're left with the simplified form: \(3\log(x) + 2\log(y)\). This simplification not only makes it easier to work with the expression but also provides a clearer insight into its components. Simplifying logarithms is all about methodically applying these rules for a clearer, simpler result.