Problem 1 - Contemporary Precalculus

Question

Write the given expression as a single logarithm. $$\ln x^{2}+3 \ln y$$

Step-by-Step Solution

Answer

Question: Combine the given logarithmic expression into a single logarithm: \(\ln x^2 + 3\ln y\) Answer: \(\ln(x^2y^3)\)

Step 1: Apply the Power Rule to the first term

First, we apply the Power Rule to the first term \(\ln x^2\). We have: $$\ln x^2 = 2\ln x$$ So now the expression becomes: $$2\ln x + 3\ln y$$

Step 2: Apply the Product Rule

Now, we will apply the Product Rule for logarithms to combine the two terms: $$2\ln x + 3\ln y = \ln(x^2) + \ln(y^3)$$ Using the Product Rule, we get: $$\ln(x^2) + \ln(y^3) = \ln(x^2y^3)$$

Step 3: Write the final answer

The expression has now been written as a single logarithm: $$\ln x^{2}+3 \ln y = \ln(x^2y^3)$$

Key Concepts

Power Rule

The power rule for logarithms is a handy tool when simplifying expressions. It states that for any logarithm of the form \( \ln(x^b) \), you can bring the exponent \( b \) in front of the logarithm. This transforms the expression into \( b \cdot \ln x \).
This property is extremely useful as it simplifies calculations in algebra.

• Example: \( \ln(x^3) = 3 \ln(x) \)
• The power rule helps in breaking down complex logarithmic expressions into simpler, more manageable parts.

In the given exercise, applying the power rule to \( \ln(x^2) \) converts it to \( 2 \ln x \), making it easier to combine with other terms in the equation. By understanding this rule, you can effectively manage expressions involving powers inside a logarithm.

Product Rule

The product rule is another essential property of logarithms, helping simplify expressions where multiple logs are involved. It states that the sum of two logarithms with the same base can be consolidated into a single logarithm. The rule is expressed as:
\[ \ln(a) + \ln(b) = \ln(ab) \]
This is especially true for natural logarithms, which use the base \(e\).

• The product rule allows us to express the multiplication of numbers as a sum of their logarithms.
• For example, \( \ln(2) + \ln(3) = \ln(6) \).

In our exercise, after transforming \( \ln(x^2) + 3\ln(y) \) to \( \ln(x^2) + \ln(y^3) \), we use the product rule to merge these into a single logarithmic expression: \( \ln(x^2y^3) \). This makes it possible to handle complex combinations more straightforwardly.

Natural Logarithms

Natural logarithms are a specific type of logarithm that use the base \( e \) (approximately 2.718). They are denoted by \( \ln \) and are widely used in mathematical and scientific calculations because of their natural growth behavior in processes like compounding.

• The notation \( \ln \) is short for "logarithm naturalis" and specifically refers to base \( e \).
• They are particularly convenient in calculus since they differentiate and integrate cleanly.

In the exercise, all logarithms involved are natural logarithms. Using the base \( e \), we simplify the expressions following the properties mentioned.
Understanding natural logarithms is crucial since they simplify when dealing with exponential growth and decay models, making them a vital tool in fields ranging from finance to biology.