Problem 84 - A Graphical Approach to Precalculus with Limits

Question

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$-\frac{3}{4} \log _{3} 16 p^{4}-\frac{2}{3} \log _{3} 8 p^{3}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\log_3\left(\frac{1}{32p^5}\right)\).

Step 1: Apply the Power Rule

The power rule of logarithms states that \( a \log_b(x) = \log_b(x^a) \). Apply this rule separately to each logarithm in the expression:\(-\frac{3}{4} \log_3(16p^4) = \log_3((16p^4)^{-3/4})\) and \(-\frac{2}{3} \log_3(8p^3) = \log_3((8p^3)^{-2/3})\).

Step 2: Simplify Using Exponentials

Calculate the expressions inside the logarithms:\((16p^4)^{-3/4} = 16^{-3/4} \cdot (p^4)^{-3/4} = (2^4)^{-3/4} \cdot p^{-3}\). The base 2 simplifies to \( (2^{-3}) \). Therefore, this becomes \((16p^4)^{-3/4} = 2^{-3}p^{-3}\).Similarly, for \((8p^3)^{-2/3}\), we have \(8^{-2/3} \cdot (p^3)^{-2/3} = (2^3)^{-2/3} \cdot p^{-2}\). Simplifying, the base 2 gives \(2^{-2}\) and this becomes \((8p^3)^{-2/3} = 2^{-2}p^{-2}\).

Step 3: Multiply Inside of the Logarithm

Now, write each simplified expression inside a single logarithm multiplication:\(\log_3(2^{-3}p^{-3}) = \log_3(\frac{1}{8p^3})\)and \(\log_3(2^{-2}p^{-2}) = \log_3(\frac{1}{4p^2})\).

Step 4: Combine Using Product Rule

Combine the two logarithmic expressions using the product rule \( \log_b(x) + \log_b(y) = \log_b(xy) \):\(\log_3\left(\frac{1}{8p^3}\right) + \log_3\left(\frac{1}{4p^2}\right) = \log_3\left(\frac{1}{8p^3} \times \frac{1}{4p^2}\right)\).This simplifies to \(\log_3\left(\frac{1}{32p^5}\right)\).

Step 5: Rearrange Numerically

The product inside the logarithm yields \(\frac{1}{32p^5}\), therefore the expression becomes:\(-\frac{3}{4} \log_3(16p^4) - \frac{2}{3} \log_3(8p^3) = \log_3\left(\frac{1}{32p^5}\right)\).

Key Concepts

Power Rule of Logarithms

When dealing with logarithmic expressions, understanding the **Power Rule** can greatly simplify your work. Essentially, the power rule states that if you have a logarithm of a number raised to a power, you can bring the exponent out front as a multiplier. Mathematically, it is expressed as:\[ a \log_b(x) = \log_b(x^a) \]This means the coefficient in front of a logarithm can be turned into an exponent inside the logarithm. For example, transforming \(-\frac{3}{4} \log_3(16p^4)\) using this rule involves moving \(-\frac{3}{4}\) inside, so it becomes \(\log_3((16p^4)^{-3/4})\).- This can help make complex logarithmic expressions more manageable.- Use this rule to change a multiplication problem into a power problem and vice versa.- It’s particularly useful when simplifying logarithms that have coefficients.

Product Rule of Logarithms

The **Product Rule of Logarithms** is indispensable when you want to combine logarithms. It allows you to convert the sum of two logs into a single log. The rule is:\[ \log_b(x) + \log_b(y) = \log_b(xy) \]This rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. For instance, if we have two expressions such as \(\log_3\left(\frac{1}{8p^3}\right)\) and \(\log_3\left(\frac{1}{4p^2}\right)\), applying the product rule results in:\[ \log_3\left(\frac{1}{8p^3}\right) + \log_3\left(\frac{1}{4p^2}\right) = \log_3\left(\frac{1}{32p^5}\right) \]- It simplifies addition of logarithms into a single expression.- Always ensure the bases of the logarithms are the same when applying this rule.- This rule is vital for breaking down complicated expressions with multiple log terms.

Simplification of Logarithmic Expressions

Simplifying logarithmic expressions involves several steps, each using a different property of logarithms. By following each step methodically, complex logarithmic expressions can be reduced into simpler forms. Let's outline the process:1. **Apply the Power Rule**: First, convert coefficients into exponents using the power rule. This transforms coefficients outside the log into exponents inside the log. - Example: \(-\frac{3}{4} \log_3(16p^4)\) becomes \(\log_3((16p^4)^{-3/4})\).2. **Simplify with Exponents**: Calculate the expressions inside the logs by applying the exponent rules to numbers and variables separately. Convert terms like powers of numbers to simpler forms, e.g., \(16^{-3/4}\) becomes \(2^{-3}\).3. **Use the Product Rule**: Combine the results into a single logarithm, further simplifying the expression.By meticulously applying these steps, you can reframe complicated logarithmic problems into more viewable forms. This makes them easier to interpret and solve.