Problem 46 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to combine the expression. $$\log 12+\frac{1}{2} \log 7-\log 2$$

Step-by-Step Solution

Answer

The expression combines to \( \log(6\sqrt{7}) \).

Step 1: Recognize the Logarithm Rules

Before combining the expression, identify the logarithm rules. There are three important logarithm properties we will use: the product rule \( \log_a b + \log_a c = \log_a (bc) \), the power rule \( k \log_a b = \log_a (b^k) \), and the quotient rule \( \log_a b - \log_a c = \log_a \left(\frac{b}{c}\right) \).

Step 2: Apply the Power Rule

Notice the term \( \frac{1}{2} \log 7 \). According to the power rule, \( \frac{1}{2} \log 7 = \log(7^{\frac{1}{2}}) \). Simplify this to \( \log(\sqrt{7}) \).

Step 3: Apply the Product Rule

Now combine \( \log 12 \) and \( \log(\sqrt{7}) \) using the product rule: \( \log 12 + \log(\sqrt{7}) = \log(12 \cdot \sqrt{7}) \).

Step 4: Apply the Quotient Rule

Combine \( \log(12 \cdot \sqrt{7}) \) and \( \log 2 \) using the quotient rule. We get \( \log(12 \cdot \sqrt{7}) - \log 2 = \log \left(\frac{12 \cdot \sqrt{7}}{2}\right) \).

Step 5: Simplify the Expression

Finally, simplify the expression \( \frac{12 \cdot \sqrt{7}}{2} \). This simplifies to \( 6 \sqrt{7} \), so the combined logarithmic expression is \( \log(6 \sqrt{7}) \).

Key Concepts

Product Rule

The product rule for logarithms is a helpful tool that allows you to combine two logarithms into one. Here is how it works:

• If you have two logarithms with the same base, like \( \log_a b + \log_a c \), you can combine them using the product rule.
• The product rule states that the sum of logarithms is the logarithm of their product: \( \log_a b + \log_a c = \log_a (bc) \).

It's like taking two separate pieces and sticking them together to make a new, single piece. This property is useful when you want to simplify expressions or solve logarithmic equations.
In our original exercise, we used the product rule to combine \( \log 12 \) and \( \log(\sqrt{7}) \). This resulted in \( \log(12 \cdot \sqrt{7}) \), making the expression easier to work with. The product rule is a straightforward yet powerful tool in algebra. It is one of the foundational laws that makes manipulating logarithms more manageable.

Power Rule

The power rule for logarithms allows you to bring exponents within a logarithm out in front as a coefficient. It's like a magnifying glass for the power.

• If you have a logarithm in the form of \( k \log_a b \), the power rule helps rewrite it as \( \log_a (b^k) \).
• This allows you to deal with powers and roots within logarithms more easily.

Consider the term \( \frac{1}{2} \log 7 \) in our example. Using the power rule, we transform it into \( \log(7^{1/2}) \), which simplifies further to \( \log(\sqrt{7}) \).
The power rule acts like a mathematical transformer that simplifies the expression while keeping the base and the power intact. It is particularly useful when dealing with fractional coefficients or when you need to simplify complex logarithmic expressions. By applying the power rule, we took a seemingly complex statement and made it simpler to work with.

Quotient Rule

The quotient rule for logarithms helps you combine logarithms by expressing them as the logarithm of a quotient. This is crucial when you subtract one logarithm from another.

• When you have two logarithms like \( \log_a b - \log_a c \), you can use the quotient rule.
• The quotient rule states that the difference of two logarithms is the logarithm of the division: \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \).

In our original exercise, after applying the product rule, we had \( \log(12 \cdot \sqrt{7}) \) and \( \log 2 \). Using the quotient rule, these become \( \log \left( \frac{12 \cdot \sqrt{7}}{2} \right) \).
By using the quotient rule, we can simplify expressions that involve subtraction of logarithms. It's an essential tool when you have one quantity divided by another. In our expression, this resulted in a more manageable form, \( \log(6 \sqrt{7}) \), which eases further calculations and interpretation.