Problem 275 - Calculus Volume 1

Question

Use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. \(\ln \left(\frac{6}{\sqrt{e^{3}}}\right)\)

Step-by-Step Solution

Answer

\( \ln \left( \frac{6}{\underline{\phantom{xx}}}\) can be rewritten as \( \ln(6) - \frac{3}{2} \).

Step 1: Rewrite the Square Root as an Exponent

Recognize that the square root can be expressed as an exponent: \( \sqrt{e^3} = (e^3)^{1/2} = e^{3/2} \). Substitute this into the original expression to get \( \ln \left( \frac{6}{e^{3/2}} \right) \).

Step 2: Apply the Quotient Rule

Use the logarithm quotient rule, \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \), to separate the numerator and denominator: \( \ln(6) - \ln(e^{3/2}) \).

Step 3: Simplify the Logarithm of an Exponent

Use the power rule of logarithms, \( \ln(a^b) = b \cdot \ln(a) \), to simplify \( \ln(e^{3/2}) = \frac{3}{2} \ln(e) \). Recall that \( \ln(e) = 1 \), so this expression becomes \( \frac{3}{2} \).

Step 4: Combine the Expressions

Combine the expressions from Step 2 and Step 3: \( \ln(6) - \frac{3}{2} \).

Key Concepts

Logarithm Rules

Logarithm rules are essential tools for manipulating logarithmic expressions. These rules allow you to break down complex logarithmic problems into simpler parts. Some of the most important rules are:

• Product Rule: \( \log_b(m \times n) = \log_b(m) + \log_b(n) \)
• Quotient Rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
• Power Rule: \( \log_b(m^n) = n \times \log_b(m) \)

In our exercise, you can see how these rules are applied. We used the Quotient Rule to separate the numerator and the denominator, which led to two simpler logarithms. Then, with the Power Rule, we simplified the expression with an exponent to further simplify the problem. By recognizing and applying these rules, you can tackle logarithmic expressions with more confidence.

Exponential Expressions

Exponential expressions are often encountered in mathematical problems and relate to terms with a base raised to a power. These can initially seem difficult, but understanding how they work simplifies their manipulation, especially when dealing with logarithms.For example, \( e^3 \) is an exponential expression, where \( e \) is the base and \( 3 \) is the exponent. A square root, like \( \sqrt{e^3} \), can also be expressed exponentially. It becomes \( (e^3)^{1/2} \), which further simplifies to \( e^{3/2} \). Recognizing these forms makes it easier to apply logarithm rules, as we've done in the exercise. Keep in mind that converting roots to exponents is a handy trick for simplifying expressions when dealing with logarithms.

Logarithm Simplification

Logarithm simplification involves reducing complex logarithmic expressions into more manageable forms. This process is crucial for solving equations and simplifying mathematical problems.In the given solution, after applying the logarithm rules, the expression \( \ln(e^{3/2}) \) was simplified using the Power Rule. Knowing that \( \ln(e) = 1 \), this expression transformed into \( \frac{3}{2} \), removing the exponential element from the logarithm.Finally, combining this simplified expression with \( \ln(6) \) gives the final simplified form: \( \ln(6) - \frac{3}{2} \). The art of simplification lies in recognizing how to apply rules effectively to reach a simpler solution. Understanding this process can significantly ease your calculations and improve problem-solving skills when working with logarithmic expressions.