Problem 73 - Elementary and Intermediate Algebra

Question

Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\ln \sqrt{e x}\)

Step-by-Step Solution

Answer

\( \ln \sqrt{e x} = \frac{1}{2} + \frac{1}{2} \ln(x) \)

Step 1: Express the square root as an exponent

Recall that a square root can be expressed as an exponent. Hence, \( \sqrt{e x} = (e x)^{1/2} \). Substitute this into the original expression: \( \ln((e x)^{1/2}) \).

Step 2: Apply the Power Rule for Logarithms

The Power Rule states that \( \ln(a^b) = b \cdot \ln(a) \). Using this rule, we have: \( \ln((e x)^{1/2}) = \frac{1}{2} \cdot \ln(e x) \).

Step 3: Apply the Product Rule for Logarithms

The Product Rule says \( \ln(a b) = \ln(a) + \ln(b) \). Apply this rule to \( \ln(e x) \): \( \ln(e x) = \ln(e) + \ln(x) \).

Step 4: Simplify Using Known Values

Since \( \ln(e) = 1 \), substitute into the expression: \( \ln(e) + \ln(x) = 1 + \ln(x) \).

Step 5: Final Expression Using Distributive Property

Distribute \( \frac{1}{2} \) over the expression from Step 4: \( \frac{1}{2} \cdot (1 + \ln(x)) = \frac{1}{2} + \frac{1}{2} \ln(x) \). Thus, the expression \( \ln \sqrt{e x} \) is written as the sum: \( \frac{1}{2} + \frac{1}{2} \ln(x) \).

Key Concepts

Logarithm Rules

Logarithm rules are essential when working with logarithmic expressions. They allow us to transform expressions into simpler forms and solve complex problems more efficiently. The primary rules of logarithms include:

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \). This rule breaks down the logarithm of a product into the sum of individual logarithms.
• Quotient Rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \). This helps express the logarithm of a division as a difference.
• Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \). It allows us to pull the exponent in front of the logarithm as a multiplier.

Understanding these rules provides a strong foundation for simplifying logarithmic expressions and solving equations involving logarithms.

Power Rule

The power rule for logarithms is a critical tool in simplifying expressions. It states that the logarithm of a power, \( \log_b(m^n) \), can be rewritten as \( n \cdot \log_b(m) \). This rule is particularly useful when dealing with expressions with exponents.
For example, if you have \( \ln((e x)^{1/2}) \), you can apply the power rule to bring the exponent \( \frac{1}{2} \) down in front of the logarithm. This simplifies the expression significantly, turning it into \( \frac{1}{2} \cdot \ln(e x) \).

By using the power rule, complex expressions become easier to handle, allowing us to identify further simplifications and evaluate the expression more straightforwardly.

Product Rule

Another essential logarithm rule is the product rule. It states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, it is expressed as \( \ln(a b) = \ln(a) + \ln(b) \).
In the given exercise, we have the expression \( \ln(e x) \), which is a product of \( e \) and \( x \). By applying the product rule, we can split this into two separate logarithmic terms: \( \ln(e) \) and \( \ln(x) \).
Remember, \( \ln(e) \) is a well-known constant equal to 1. This simplification is a key step that enables us to break complex expressions into simpler parts before applying other rules or finalizing the expression. It essentially Untangles the logarithms, making further operations easier.

Logarithmic Simplification

Logarithmic simplification involves using rules to make expressions more manageable. It combines strategies such as applying known values, distributing factors, and recognizing simplifications.
In our example, after applying the power and product rules, we are left with \( \frac{1}{2}(1 + \ln(x)) \). Here, \( \ln(e) \) simplifies to 1, an essential step in shortening the equation. Next, using the distributive property allows us to simplify further:

• Distribute: \( \frac{1}{2} \cdot 1 \) gives \( \frac{1}{2} \)
• Continue distributing: \( \frac{1}{2} \cdot \ln(x) \) turns into \( \frac{1}{2} \ln(x) \)

Combining these results produces the final simplified expression: \( \frac{1}{2} + \frac{1}{2} \ln(x) \).
This final step in simplification is crucial for expressing logarithmic terms in their simplest form, making calculations easier and solutions clearer.