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Chapter 4: Problem 2

Simplify the following expressions. \(\ln x^{5}-\ln x^{3}\)

Short Answer

Expert verified
2 \ln x

Step by step solution

01

Apply the Logarithm Power Rule

Use the power rule for logarithms, which is \(\ln a^b = b \ln a\). Apply this rule to each logarithmic term: \(\ln x^5 = 5 \ln x\) and \(\ln x^3 = 3 \ln x\).
02

Substitute Back into the Expression

Replace \(\ln x^5\) and \(\ln x^3\) in the original expression with the simplified forms: \(5 \ln x - 3 \ln x\).
03

Combine Like Terms

Since both terms have \(\ln x\) as a common factor, combine the coefficients: \(5 \ln x - 3 \ln x = (5 - 3) \ln x = 2 \ln x\).

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Key Concepts

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

logarithm power rule
The logarithm power rule is a useful property that allows you to simplify logarithmic expressions. It states that for any positive number, for example in the expression \(\text{ln}(a^b) = b \text{ln}(a)\), you can move the exponent in front of the logarithm as a multiplier.

This rule makes working with logarithms simpler because it converts the exponentiation inside the logarithm into multiplication outside the logarithm.

In our example, we start with \(\text{ln}(x^5) - \text{ln}(x^3)\). Using the power rule, we rewrite these as \(\text{ln}(x^5) = 5 \text{ln}(x)\) and \(\text{ln}(x^3) = 3 \text{ln}(x)\).
combining like terms
Combining like terms is a fundamental concept in algebra that also applies to logarithmic expressions to simplify them. When terms have the same variable or, in this case, the same logarithmic base, we can combine their coefficients.

For instance, in the equation \(\text{ln}(x^5) - \text{ln}(x^3)\), we reduced this to \(\text{5 \text{ln}(x) - 3 \text{ln}(x)}\)

The next step is to recognize that both terms include \(\text{ln}(x)\) as a common factor. Thus, we can combine the coefficients (5 and 3) by subtracting them: \(5 \text{ln}(x) - 3 \text{ln}(x) = (5-3) \text{ln}(x)\)

This simplifies to \((2 \text{ln}(x))\).
logarithmic properties
Logarithms have several important properties that help simplify complex expressions.

Some key properties include:
  • **Product Rule:** \(\text{ln}(a \times b) = \text{ln}(a) + \text{ln}(b)\)
  • **Quotient Rule:** \(\text{ln}(a/b) = \text{ln}(a) - \text{ln}(b)\)
  • **Power Rule:** \(\text{ln}(a^b) = b \text{ln}(a)\)
In our example, we used the power rule to manage the exponents. Then, by combining like terms (common factors), we simplified the expression.

In summary, understanding these properties allows you to break down and simplify complex logarithmic expressions easily.

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