Problem 29 - Algebra 2

Question

Expand each logarithm. \(\log \frac{s \sqrt{7}}{t^{2}}\)

Step-by-Step Solution

Answer

The expanded form of the given logarithm is \[ \log s + \frac{1}{2}\log 7 - 2\log t \]

Step 1: Apply the quotient rule

The quotient rule of logarithms states that the logarithm of a quotient is the difference of the logarithms. So, the expression can be rewritten as: \[ \log s \sqrt{7} - \log t^{2} \]

Step 2: Apply the product rule

The product rule of logarithms states that the logarithm of a product is the sum of the logarithms. Apply this rule to the first part of the expression: \[ \log s + \log \sqrt{7} - \log t^{2} \]

Step 3: Apply the power rule

The power rule of logarithms states that the logarithm of a power is the product of the exponent and the logarithm. Apply this rule to reduce the square root and square in the expression: \[ \log s + \frac{1}{2}\log 7 - 2\log t \]

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a powerful tool for simplifying expressions involving logs. According to this rule, the logarithm of a division is equal to the difference between two individual logarithms. This means if you have a log expression like \( \log \frac{A}{B} \), you can rewrite it as \( \log A - \log B \). This is especially useful when dealing with complex fractions, as it breaks them down into simpler parts.

In practice, this rule helps you separate the numerator and the denominator into distinct logarithmic components, making it easier to manipulate or solve. For example, given the expression \( \log \frac{s \sqrt{7}}{t^{2}} \), applying the quotient rule simplifies this to \( \log s \sqrt{7} - \log t^{2} \). By breaking the expression down, we make it more digestible and set it up for further simplification with other logarithmic rules.

Product Rule of Logarithms

The product rule of logarithms allows us to break down expressions involving multiplication inside a logarithm. Using this rule, the logarithm of a product is the same as the sum of the logarithms of the factors. So, if you're working with \( \log(A \times B) \), you can rewrite it as \( \log A + \log B \).

This rule facilitates expanding logarithmic expressions by separating multiplied terms into individual logarithms. In our example, after applying the quotient rule, we have \( \log s \sqrt{7} \). Using the product rule further, this becomes \( \log s + \log \sqrt{7} \). This separation simplifies the solving process, especially when dealing with multiple compounded terms.

Breaking down logs into simpler terms can illuminate patterns or simplify further calculations, making logarithmic operations more straightforward and manageable.

Power Rule of Logarithms

The power rule of logarithms helps us handle powers inside log expressions. It states that the logarithm of a term raised to a power can be rewritten by multiplying the exponent by the logarithm of the base. In other words, \( \log(A^n) = n \cdot \log A \).

This rule is particularly useful when dealing with squares, cubes, or roots in logarithmic expressions. For instance, in our expression after applying the product rule, we have \( \log \sqrt{7} \) and \( \log t^{2} \). Applying the power rule gives us \( \frac{1}{2} \log 7 \) for the square root (since \( \sqrt{7} \) is the same as \( 7^{0.5} \)) and \( 2 \log t \) for the square.

This method allows us to streamline logarithmic expressions by effectively removing exponents, making the entire expression easier to handle and interpret. Understanding this rule is essential for efficiently solving complex logarithmic equations.