Problem 50 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to combine the expression. $$\ln (a+b)+\ln (a-b)-2 \ln c$$

Step-by-Step Solution

Answer

\(\ln\left(\frac{a^2-b^2}{c^2}\right)\)

Step 1: Apply the Product Rule

The product rule states that the sum of two logarithms is equal to the logarithm of the product of the arguments. We use this rule to combine \(\ln(a+b)\) and \(\ln(a-b)\):\[\ln(a+b) + \ln(a-b) = \ln((a+b)(a-b)).\]

Step 2: Simplify the Expression Inside the Logarithm

Applying the identity for the product of two binomials, \((a+b)(a-b) = a^2 - b^2\), we get:\[\ln((a+b)(a-b)) = \ln(a^2 - b^2).\]

Step 3: Apply the Power Rule

The power rule for logarithms states that \(b \ln(a) = \ln(a^b)\). We apply this rule to \(-2 \ln(c)\):\[-2 \ln(c) = \ln(c^{-2}).\]

Step 4: Combine Using the Quotient Rule

Using the quotient rule, \(\ln(A) - \ln(B) = \ln\left(\frac{A}{B}\right)\), combine \(\ln(a^2-b^2)\) and \(-2\ln(c)\):\[\ln(a^2-b^2) - \ln(c^2) = \ln\left(\frac{a^2-b^2}{c^2}\right).\]

Step 5: Final Expression

The combined expression using the laws of logarithms is \(\ln\left(\frac{a^2-b^2}{c^2}\right)\).

Key Concepts

Product Rule

The product rule is a fundamental law of logarithms that helps us simplify expressions with multiple logarithms. It's particularly useful when you encounter a sum of two logarithms, as in our exercise. Essentially, the product rule states:

• When you add two logarithmic expressions, it is equivalent to the logarithm of the product of their arguments.

For example, if you have \( \ln(x) + \ln(y) \),it combines to \( \ln(xy) \).The key takeaway is that with the product rule, the operation moves inside the logarithm, transforming addition into multiplication.
This means when you pair up logarithms, you can think of it as creating a single log expression by taking the product of the original arguments.

Power Rule

The power rule is another essential tool in the arsenal of logarithmic laws. It's particularly used when a logarithm is multiplied by a constant. The power rule states:

• When you have a constant multiplied by a logarithm, \( b \ln(a) \),this can be rewritten as \( \ln(a^b) \).

A good example is transforming \( -2 \ln(c) \)into \( \ln(c^{-2}) \).Here, the constant \(-2\)becomes the exponent of the base \(c\).So, instead of scaling the logarithm output, you scale the input, simplifying expressions significantly.
It’s about essentially flipping a multiplication outside the log into an exponentiation inside.

Quotient Rule

The quotient rule is invaluable when dealing with subtraction of logarithms. This rule assists in combining logarithmic terms by transforming subtraction into division.According to the quotient rule:

• A difference of logarithms, \( \ln(A) - \ln(B) \),turns into \( \ln\left(\frac{A}{B}\right) \).

In practice, consider our example of \( \ln(a^2-b^2) - \ln(c^2) \),which simplifies to \( \ln\left(\frac{a^2-b^2}{c^2}\right) \).The elegant aspect of the quotient rule is how subtraction outside becomes division within the logarithm, effectively combining two logs into a single one.
It makes simplifying expressions far more manageable, often collapsing complex differences into neat fractions.

Combining Logarithmic Expressions

Combining logarithmic expressions involves using the laws of logarithms to reduce several log terms into one. In the specific problem we tackled, we combined all the rules we've talked about: the product, power, and quotient rules.Here's a quick overview:

• Use the Product Rule: Additive logarithms \( \ln(a+b) + \ln(a-b) \)transform into \( \ln((a+b)(a-b)) \).
• Employ the Power Rule: \(-2\ln(c)\)translates to \( \ln(c^{-2}) \).
• Apply the Quotient Rule: Combine results into \( \ln\left(\frac{a^2-b^2}{c^2}\right) \).

By systematically using these rules, you can overhaul a complex expression into a tidy logarithm.
Combining logarithmic expressions means mastering these rules for elegant and effective simplifications.