Problem 96 - College Algebra and Calculus: An Applied Approach

Question

Condense the expression to the logarithm of a single quantity.\(\frac{3}{2} \ln (z-2)+\ln z$$\frac{3}{2} \ln (z-2)+\ln z\)

Step-by-Step Solution

Answer

The expression condensed to a single quantity is \( \ln (z(z-2)^{3/2}) \)

Step 1: Apply Power Rule

Initiate by applying the power rule of logarithms. This rule dictates that a coefficient can be moved to the exponent of the argument. Hence, rewrite the expression as follows: \( \ln ((z-2)^{3/2}) + \ln z \). It specifically applied to the first term. Now the expression has two logarithms being added.

Step 2: Apply Product Rule

Now, it’s time to apply the product rule of logarithms which states that the sum of two logarithms with the same base equals the logarithm of the product of their arguments. Hence, rewrite the expression as: \( \ln ((z-2)^{3/2} * z) \).

Step 3: Simplify the Expression

Now simplify the current expression to wrap it up. The expression is given by: \( \ln (z(z-2)^{3/2}) \). This is the logarithm of a single quantity as requested.

Key Concepts

Power Rule

The Power Rule of logarithms is a handy tool. It tells us that when you have a coefficient in front of a logarithm, you can "move" it to be an exponent of the argument instead. Think of it as bouncing the number up! Here's how it works: if you have an expression like \( a \log_b(x) \), you can rewrite it as \( \log_b(x^a) \). This makes expressions cleaner and simpler.
For example, in the expression \( \frac{3}{2} \ln(z-2) \), the \( \frac{3}{2} \) in front gets flipped into \( (z-2)^{3/2} \) inside the logarithm. It's like shifting places but changes the form. Remember this rule for when you need to manipulate expressions; it makes them easier to handle and solve!

Product Rule

The Product Rule is super useful when dealing with logarithms that are being added together. It states that when you add the logs of two numbers with the same base, you can combine them into a single log of their product. Here's what it looks like
\( \log_b(x) + \log_b(y) = \log_b(xy) \). This trick works in reverse too, which can be just as helpful.
Consider our expression \( \ln((z-2)^{3/2}) + \ln(z) \), where both logs have the same base. You can merge them to a neat single log: \( \ln((z-2)^{3/2} \cdot z) \). With the Product Rule, you simplify the task of working with multiple logs by rolling them into one!

Simplifying Expressions

Simplifying is the art of making expressions more compact and easier to understand. After applying rules like the Power Rule and Product Rule, the next phase is to tidy up the expression into its simplest form. This step should lead to the clearest representation of the solution, which is particularly helpful in exams or assignments.
Looking at our worked expression \( \ln((z-2)^{3/2} \cdot z) \), the objective was to express it as the logarithm of a single quantity. By retaining the expression in this concise and neat form \( \ln(z(z-2)^{3/2}) \), we achieve full simplification. In mathematical communications, a simplified expression not only saves space but also reduces potential errors in computing with complex multi-step problems.