Problem 83

Question

Simplify the following expressions: (a) $\ln x^{2}+\ln x^{3}$ (b) $\ln x^{4}-\ln x^{-2}$ (c) $\ln \left(x^{2}-1\right)-\ln (x+1)$ (d) $\ln x^{-1}+\ln x^{-3}$

Step-by-Step Solution

Verified

Answer

(a) $ \ln(x^{5}) $; (b) $ \ln(x^{6}) $; (c) $ \ln(x - 1) $; (d) $ -4\ln(x) $.

1Step 1: Apply the Product Rule

For part (a), apply the product rule of logarithms: $ \ln a + \ln b = \ln(ab) $. Here, $ a = x^2 $ and $ b = x^3 $. Combine to get $ \ln(x^{2}\cdot x^{3}) = \ln(x^{5}) $.

2Step 2: Apply the Quotient Rule

For part (b), apply the quotient rule of logarithms: $ \ln a - \ln b = \ln\left(\frac{a}{b}\right) $. Here, $ a = x^4 $ and $ b = x^{-2} $. Combine to get $ \ln\left(\frac{x^{4}}{x^{-2}}\right) = \ln(x^{6}) $.

3Step 3: Simplify Using Basic Algebra

For part (b) further simplification, note that dividing by a negative exponent is the same as multiplying: $ x^{4}\cdot x^{2} $. Therefore, simplified form is still $ x^{6} $.

4Step 4: Apply the Quotient Rule Again

For part (c), apply the quotient rule: Here, $ a = x^2 - 1 $ and $ b = x+1 $. Combine to get $ \ln\left( \frac{x^2 - 1}{x + 1} \right) $.

5Step 5: Simplify the Rational Expression

The expression $ \frac{x^2 - 1}{x + 1} $ can be simplified using algebra (difference of squares): $ x^2 - 1 = (x - 1)(x + 1) $. Therefore, it becomes $ \frac{(x - 1)(x + 1)}{x + 1} = x - 1 $. Thus, the simplified form is $ \ln(x - 1) $.

6Step 6: Apply the Product Rule Again

For part (d), apply the product rule: Here, $ a = x^{-1} $ and $ b = x^{-3} $. Combine to get $ \ln(x^{-1}\cdot x^{-3}) = \ln(x^{-4}) $.

7Step 7: Simplify Using Exponent Rules

Note that $ x^{-4} = \frac{1}{x^{4}} $. Therefore, the logarithm expression is $ \ln\left(\frac{1}{x^{4}}\right) $, which simplifies to $ -4\ln(x) $ using the logarithm power rule.

Key Concepts

Product RuleQuotient RuleSimplify ExpressionsExponent Rules

Product Rule

The product rule for logarithms is a handy tool when you're dealing with two logarithmic expressions added together. It states that the sum of the logarithms of two numbers is equal to the logarithm of their product. So if you have

$ \ln a + \ln b = \ln(ab) $

This rule helps to consolidate expressions and is useful to simplify complex equations involving logs.
For example, in $ \ln x^2 + \ln x^3 $, you can combine terms into a single logarithmic expression:

$ \ln(x^{2} \cdot x^{3}) = \ln(x^{5}) $

Quotient Rule

The quotient rule for logarithms deals with the difference between two logarithms, showing it as the logarithm of their quotient. The formula looks like this:

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

This rule is helpful to simplify expressions where subtraction is involved.
For instance, in $ \ln x^4 - \ln x^{-2} $, one can use the quotient rule to merge them into:

$ \ln\left(\frac{x^{4}}{x^{-2}}\right) = \ln(x^{6}) $

Simplify Expressions

Simplifying expressions in logarithms often requires using basic algebraic techniques. For example, some expressions need factoring or expanding.
Let's consider the expression: $ \ln\left( \frac{x^2 - 1}{x + 1} \right) $ from the example given above. This expression simplifies with algebraic factoring.
Recognize $ x^2 - 1 $ as a difference of squares, which factors to:

$ (x-1)(x+1) $

Then simplify the rational expression

$ \frac{(x-1)(x+1)}{x+1} = x-1 $

Thus, the original log simplifies to

$ \ln(x-1) $

Exponent Rules

Understanding exponent rules is crucial when dealing with logarithms because they frequently involve powers.
For example, understanding how to manage negative exponents simplifies complex expressions.
In the case of $ \ln x^{-1} + \ln x^{-3} $, the product rule allows us to simplify by adding the powers of $ x $: