Problem 70 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\( \ln \sqrt{x y} = \frac{1}{2} \ln(x) + \frac{1}{2} \ln(y) \)

Step 1: Convert the Square Root to Exponent

The expression \( \ln \sqrt{x y} \) can initially be rewritten by expressing the square root as an exponent: \( \ln (x y)^{1/2} \). This is because the square root of a term is equivalent to raising it to the power of \( \frac{1}{2} \).

Step 2: Apply the Power Rule for Logarithms

Using the power rule, which states \( \ln(a^b) = b \cdot \ln(a) \), we can rewrite \( \ln (x y)^{1/2} \) as \( \frac{1}{2} \cdot \ln(x y) \).

Step 3: Apply the Product Rule for Logarithms

The product rule states that \( \ln(a b) = \ln(a) + \ln(b) \). Using this, we can split \( \ln(x y) \) into \( \ln(x) + \ln(y) \).

Step 4: Combine All Steps

Substituting the result from the product rule into the expression from Step 2, we get: \( \frac{1}{2} \cdot (\ln(x) + \ln(y)) \). Distributing the \( \frac{1}{2} \) gives the final expression: \( \frac{1}{2} \ln(x) + \frac{1}{2} \ln(y) \).

Key Concepts

Logarithm Rules

Logarithm rules are fundamental tools for simplifying or transforming logarithmic expressions. They help break down complex logarithms into more manageable parts and often allow the expression of these logs as sums or differences. Here are some basic rules:

• The Product Rule states that for any positive numbers \(a\) and \(b\), \( \log(ab) = \log(a) + \log(b) \).
• The Quotient Rule provides that \( \log(\frac{a}{b}) = \log(a) - \log(b) \) for any positive numbers \(a\) and \(b\).
• The Power Rule lets us express \( \log(a^b) \) as \( b \cdot \log(a) \).

These rules make it easier to manipulate and calculate logarithms, especially when dealing with more complex algebraic expressions in calculus or algebra.
They are vital for solving logarithmic equations by isolating the variable or simplifying expressions before computation.

Square Root as Exponent

Square roots can be tricky in calculations, but they can be made much simpler by expressing them as exponents. The square root of a number \(x\), denoted as \(\sqrt{x}\), is equivalent to \(x^{1/2}\).
This transformation is extremely useful in logarithmic expressions. It allows us to utilize the Power Rule of logarithms, turning complex root expressions into simpler ones. For example, if you have \(\ln \sqrt{x}\), you can rewrite this as \(\ln(x^{1/2})\).
By turning square roots into exponents, we maintain consistency across mathematical expressions and can easily apply additional logarithmic rules. This substitution often brings clarity and ease to solving logarithmic problems.

Product Rule for Logarithms

The Product Rule for logarithms is a critical concept that simplifies expressions involving multiplication. According to this rule, the logarithm of a product is equivalent to the sum of the logarithms of the individual factors:

• \( \ln(ab) = \ln(a) + \ln(b) \)

This rule is especially powerful when you are trying to break down complex logarithmic expressions into simpler terms that can be easily handled or further manipulated.
For example, if you encounter \(\ln(xy)\), you can apply this rule to separate it into \(\ln(x) + \ln(y)\).
Using the Product Rule helps break apart products inside logarithms, helping to simplify calculations and make complex problems easier to solve and understand.

Power Rule for Logarithms

The Power Rule of logarithms is perhaps one of the most versatile tools you will use. It allows you to move exponents in and out of logarithmic expressions, simplifying them greatly. Mathematically, it is expressed as:

• \( \ln(a^b) = b \cdot \ln(a) \)

This rule is especially handy for expressions involving powers, as it transforms exponents into multipliers outside the natural logarithm.
In practice, if you have \( \ln(x^{1/2}) \), using the Power Rule, this becomes \( \frac{1}{2} \cdot \ln(x) \).
This conversion is not only crucial for simplifying logarithmic equations but also aids in integrating and differentiating logarithmic functions in calculus. With this rule, you maintain simplicity and consistency across complex expressions.