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Chapter 5: Problem 49

Express as a single logarithm and, if possible, simplify. $$\ln x-3[\ln (x-5)+\ln (x+5)]$$

Short Answer

Expert verified
\(\text{ln} \frac{x}{[(x-5)(x+5)]^3}\).

Step by step solution

01

Apply Logarithm Properties

Use the property of logarithms that states \(\text{log}(a) + \text{log}(b) = \text{log}(ab)\). Apply this to \(\text{ln}(x-5) + \text{ln}(x+5)\): \(\text{ln}((x-5)(x+5))\).
02

Distribute the Coefficient

Distribute the coefficient \(-3\) to the logarithm: \(\text{ln} x - 3 \text{ln}((x-5)(x+5))\).
03

Apply Logarithm Power Rule

Utilize the power rule of logarithms: \(\text{log}(a^b) = b\text{log}(a)\). Apply this to \(-3 \text{ln}((x-5)(x+5))\), converting it to \(\text{ln}((x-5)(x+5))^{-3}\).
04

Combine Logarithm Terms

Combine the logarithm terms using subtraction: \(\text{ln}(x) - \text{ln}((x-5)(x+5))^{-3}\) becomes \(\text{ln} \frac{x}{[(x-5)(x+5)]^3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Properties
Logarithm properties are crucial tools that help simplify and manipulate logarithmic expressions. One important property is the product rule.

The product rule states that the logarithm of a product is equal to the sum of the logarithms. In mathematical terms, \(\text{log}(a) + \text{log}(b) = \text{log}(ab)\).

In our example, we started with \(\text{ln}(x-5) + \text{ln}(x+5)\). By applying the product rule, we combined these into \(\text{ln}((x-5)(x+5))\).

This step is key in simplifying logarithmic expressions, especially when dealing with multiple terms and coefficients.
Logarithm Power Rule
The logarithm power rule is another essential tool. This rule states that the logarithm of a number raised to a power is equal to the exponent times the logarithm of the base.

Mathematically, this is written as \(\text{log}(a^b) = b\text{log}(a)\).

In our exercise, we had the term \(-3 \text{ln}((x-5)(x+5))\).

Using the power rule, we converted it to \(\text{ln}((x-5)(x+5))^{-3}\).

This step allows us to move the exponent inside the logarithm, which is helpful for combining terms.
Logarithm of a Product
The property of the logarithm of a product is utilized often when simplifying expressions. Remember, \(\text{log}(a) + \text{log}(b) = \text{log}(ab)\).

This principle helps us transform sums of logarithms into a single logarithm of a product.

In our specific problem, we used this property twice: first to combine \(\text{ln}(x-5) + \text{ln}(x+5)\) into \(\text{ln}((x-5)(x+5))\), and then in the final step to combine \(\text{ln}(x) - \text{ln}((x-5)(x+5))^{-3}\) into a single logarithm \(\text{ln}\frac{x}{((x-5)(x+5))^3}\).

This final form is often more compact and easier to work with in further calculations or applications.

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