Problem 80 - Precalculus with Limits

Question

Condensing a Logarithmic Expression In Exercises \(67-82,\) condense the expression to the logarithm of a single quantity. 2$$[3 \ln x-\ln (x+1)-\ln (x-1)]$$

Step-by-Step Solution

Answer

The condensed form of the expression is: \(\ln ((x^3 / (x^2 - 1))^2)\)

Step 1: Combine Logarithms Using the Product Rule and Quotient Rule

Rewrite the given expression separating minus parts from each other. We get: \(2[3 \ln x - (\ln (x+1) + \ln(x-1))]\). After this, apply logarithm rules. Using the Product Rule, we see that \( \ln a - \ln b = \ln (a/b) \). Therefore, \(\ln (x+1) + \ln(x-1) = \ln (x+1)(x-1)\). Thus, the expression becomes \(2[3 \ln x - \ln (x^2 - 1)]\).

Step 2: Use the Power Rule of Logarithms

The Power Rule of logarithms states that the coefficient of logarithm can be moved as the power of the subject of the logarithm. Therefore, the expression becomes \(2[\ln x^3 - \ln (x^2 - 1)]\).

Step 3: Apply the Quotient Rule

Now that we have only two logarithms separated by a subtraction symbol, apply the Quotient Rule to combine the logarithms into one. The new expression becomes \(2\ln (x^3 / (x^2 - 1))\).

Step 4: Apply the Power Rule Again

Now, we will again use the Power Rule as in step 2 to move the coefficient of the logarithm. Hence, the expression becomes \(\ln ((x^3 / (x^2 - 1))^2)\).

Key Concepts

Product Rule

Understanding the Product Rule for logarithms can make solving complex logarithmic expressions much simpler. The Product Rule states that the sum of log terms

• \( \ln a + \ln b = \ln(ab) \)

This is useful when you need to condense multiple logarithms into one. Imagine you have two quantities multiplied together within the log function. The Product Rule allows you to take their individual logarithms and combine them into a single log expression.

In the example given, applying the Product Rule transformed two separate log terms, \( \ln(x+1) + \ln(x-1) \), using the rule \( \ln (x+1) + \ln(x-1) = \ln ((x+1)(x-1)) \). Here, the expression was simplified to \( \ln (x^2 - 1) \). This reduces the problem to one of managing fewer logarithmic terms and makes the solution more straightforward.

Quotient Rule

The Quotient Rule for logarithms is another key tool for condensing expressions. It states:

• \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \)

You can use this rule whenever you have one logarithm subtracted from another. It allows you to combine these into one log term, which can dramatically simplify equations.

In our problem, the application of the Quotient Rule arose after manually combining some logarithms. The expression \( 2[\ln x^3 - \ln (x^2 - 1)] \) was simplified using this rule to become \( 2\ln \left( \frac{x^3}{x^2 - 1} \right) \). This reduction brought the expression to a much more compact form, as you only need to deal with a single logarithm at this stage, instead of dealing with the separate components.

Power Rule

The Power Rule is a very handy rule when dealing with logarithmic expressions that involve exponents. It tells us that:

• \( n \ln a = \ln (a^n) \)

This is particularly useful for moving coefficients outside of the log function inside as exponents.

In the given example, you saw how the Power Rule was used initially with \( 3 \ln x \) to convert it to \( \ln x^3 \). Later, after arriving at \( 2\ln \left( \frac{x^3}{x^2 - 1} \right) \), the Power Rule was applied again to make the expression \( \ln \left( \frac{x^3}{x^2 - 1} \right)^2 \). This manipulation makes the expression tidy and fully condenses it to a single logarithmic form, enabling the student to work with one simpler entity.