Problem 56 - Precalculus with Calculus Previews

Question

In Problems \(51-56\), use the laws of logarithms in Theorem \(6.2 .1\) to rewrite the given expression as one logarithm. $$ 5 \ln 2+2 \ln 3-3 \ln 4 $$

Step-by-Step Solution

Answer

The expression simplifies to \( \ln 4.5 \).

Step 1: Apply the Power Rule

Start by applying the power rule of logarithms, which states that for any real number \( a \) and positive number \( M \), \( a \ln M = \ln(M^a) \). This rule allows multiplication with the logarithm to be converted into an exponent. Apply the rule to each term: \[ 5 \ln 2 = \ln(2^5), \quad 2 \ln 3 = \ln(3^2), \quad \text{and} \quad 3 \ln 4 = \ln(4^3). \] This simplifies the terms to: \[ \ln 32 + \ln 9 - \ln 64. \]

Step 2: Combine Using the Product Rule

Next, apply the product rule of logarithms, which states \( \ln A + \ln B = \ln (A \times B) \). Combine the terms \( \ln 32 + \ln 9 \) into a single logarithm by multiplying the arguments:\[ \ln 32 + \ln 9 = \ln (32 \times 9) = \ln 288. \]

Step 3: Combine Using the Quotient Rule

Now use the quotient rule of logarithms, which states \( \ln A - \ln B = \ln \left( \frac{A}{B} \right) \). Subtract \( \ln 64 \) from \( \ln 288 \) to form a single logarithm:\[ \ln 288 - \ln 64 = \ln \left( \frac{288}{64} \right). \] Simplify the fraction to find:\[ \ln \left( \frac{288}{64} \right) = \ln 4.5. \]

Key Concepts

Power Rule

The power rule of logarithms is a handy tool for manipulating logarithmic expressions. It allows you to take a constant multiplier in front of a logarithm and change it into an exponent within the logarithm. This is particularly useful for simplifying expressions. The rule states that for any real number \(a\) and positive number \(M\), \(a \ln M = \ln(M^a)\). For example, if you have an expression like \(5 \ln 2\), you can apply the power rule to transform it into \(\ln(2^5)\). This results in a more straightforward format where you now have \(\ln 32\).
Similarly, calculate \(2 \ln 3\) by adjusting it to \(\ln(3^2)\), which gives you \(\ln 9\). And for \(-3 \ln 4\), apply the power rule to convert it to \(\ln(4^3)\), which becomes \(- \ln 64\).
Utilizing the power rule helps to not only simplify your calculations but also makes the expressions cleaner and more uniform.

Product Rule

The product rule of logarithms is another powerful tool that makes working with logarithmic expressions simpler. The rule helps in combining the logarithms of multiple expressions into one. It states that the logarithm of a product is the sum of the logarithms of the factors: \(\ln A + \ln B = \ln (A \times B)\).
This can be particularly beneficial when you come across expressions with addition between logs. For example, if you have \(\ln 32 + \ln 9\), you can combine these into a single logarithmic expression by multiplying the arguments:

• \(\ln 32 + \ln 9\) becomes \(\ln (32 \times 9)\)
• \(\ln (32 \times 9)\) simplifies to \(\ln 288\)

Using the product rule allows you to consolidate expressions efficiently and can reduce complexity in your calculations.

Quotient Rule

The quotient rule of logarithms allows the transformation of subtraction between logs into a division within a single logarithm. This rule states that the logarithm of a quotient is the difference of the logarithms: \(\ln A - \ln B = \ln \left( \frac{A}{B} \right)\).
This can significantly simplify expressions where logs are subtracted. For instance, consider the expression \( \ln 288 - \ln 64 \). By using the quotient rule, you can rewrite it as a single log:

• \(\ln 288 - \ln 64\) becomes \(\ln \left( \frac{288}{64} \right)\)
• Simplifying the fraction \(\frac{288}{64}\) gives \(\frac{9}{2}\) or \(4.5\), thus \(\ln 4.5\)

Applying the quotient rule in this way helps join separate logs into one manageable expression, streamlining the solving process.