Problem 275 - Calculus Volume 1

Question

For the following exercises, 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

\( \ln(6) - \frac{3}{2} \)

Step 1: Apply the Quotient Rule for Logarithms

Using the quotient property of logarithms, \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \), split the given expression into two separate logarithms: \( \ln(6) - \ln(\sqrt{e^{3}}) \).

Step 2: Simplify the Logarithm of a Square Root

Recognize that the square root can be expressed as a power: \( \sqrt{e^{3}} = (e^{3})^{1/2} = e^{3/2} \). Thus, we can rewrite the expression from Step 1 as \( \ln(6) - \ln(e^{3/2}) \).

Step 3: Apply the Power Rule for Logarithms

Using the power property of logarithms, \( \ln(a^{b}) = b\ln(a) \), simplify the expression \( \ln(e^{3/2}) \) to \( \frac{3}{2}\ln(e) \). Note that \( \ln(e) = 1 \) since \( e \) is the base of the natural logarithm. Thus, \( \frac{3}{2}\ln(e) = \frac{3}{2} \).

Step 4: Combine the Expressions

Substitute the simplified expression back into the equation: \( \ln(6) - \frac{3}{2} \). This is the final expression for the original logarithm as a sum and/or difference.

Key Concepts

Quotient Rule for Logarithms

The quotient rule for logarithms is a handy property when dealing with the division of two quantities inside a logarithmic function. It states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. To put this into an equation, for any positive numbers \(a\) and \(b\), the rule is: \[ \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \] This property lets us break down more complex logarithmic expressions and simplify them step by step. For instance, if you come across \( \ln \left( \frac{6}{\sqrt{e^{3}}} \right) \), you can use the quotient rule to turn it into: \[ \ln(6) - \ln(\sqrt{e^{3}}) \] This format makes it easier to proceed with further simplifications. Utilizing the quotient rule is often the first step in simplifying logarithmic expressions involving division.

Power Rule for Logarithms

The power rule for logarithms is another key logarithmic property that simplifies expressions where a variable is raised to a power inside a logarithm.It allows us to move the exponent outside the logarithm as a multiplier. For any positive number \(a\) and exponent \(b\), the rule goes like this: \[ \ln(a^b) = b \ln(a) \] This property is incredibly useful because it transforms complex expressions into simpler, more manageable ones. In our exercise, after applying the quotient rule, we were left with: \[ \ln(e^{3/2}) \] Applying the power rule here, we extract the \(3/2\) from the power, turning it into: \[ \frac{3}{2} \ln(e) \] And since \(\ln(e) = 1\), it simplifies further to just \(\frac{3}{2}\). By using the power rule, you can always make logarithmic equations easier to handle.

Natural Logarithms

Natural logarithms are a special type of logarithm that use Euler's number \(e\) as the base. Represented as \(\ln(x)\), the natural logarithm is widely used in calculus and higher mathematics due to its unique properties. Euler's number \(e\) is approximately 2.71828. One of the most important features of natural logarithms is that \(\ln(e) = 1\). This is because a logarithm tells us the power to which the base must be raised to get a certain number. And since \(e^1 = e\), the natural logarithm of \(e\) results in one. Natural logarithms often appear in problems involving exponential growth or decay. In this exercise, we saw how \(\ln(e^{3/2})\) simplifies, using this property to transform into a much simpler \(\frac{3}{2}\). Understanding natural logarithms will massively aid in working through calculus problems and beyond, providing a pathway to solving exponential equations.