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Chapter 6: Problem 25

Condense the logarithmic expression. $$6 \ln x+4 \ln y$$

Short Answer

Expert verified
The condensed form of the given logarithmic expression is \(\ln(x^6y^4)\)

Step by step solution

01

Apply the Power Rule

Rewrite the expression \(6 \ln x+4 \ln y\) using the power rule for logarithms. This transforms to \(\ln(x^6) + \ln(y^4)\)
02

Apply the Addition Rule

Now, use the addition rule for logarithms. This condenses the expression to \(\ln(x^6 \times y^4)\)

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Key Concepts

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

Power Rule for Logarithms
The power rule for logarithms is a handy tool to simplify expressions with logarithms. It allows us to move a coefficient from in front of the logarithm to an exponent inside the logarithm.
This can make complicated expressions much neater and easier to handle.When you see an expression like \(a \ln b\), you can rewrite it as \(\ln(b^a)\).
  • For example, \(5 \ln x\) turns into \(\ln(x^5)\).
  • This is because the number in front of the logarithm, the coefficient, becomes the power or exponent of the number inside the logarithm.
The power rule is particularly useful when dealing with expressions that have multiple terms with logarithms.
By applying this rule, you transform each term individually, making it easier to eventually combine them.
Addition Rule for Logarithms
The addition rule for logarithms is crucial for condensing logarithmic expressions.
When you encounter addition between logarithms, you can combine them into a single logarithmic expression.
This rule states that the logarithm of a product is equal to the sum of logarithms of its factors:
  • If you see \(\ln(a) + \ln(b)\), it can be combined to \(\ln(a \times b)\).
Using the example from the exercise, after transforming using the power rule, we had \(\ln(x^6) + \ln(y^4)\).
The addition rule allows us to combine these into a single log expression: \(\ln(x^6 \times y^4)\).
This step simplifies our work and is a vital part of condensing expressions. It's often what turns multiple terms into one much simpler term.
Logarithmic Properties
Logarithmic properties are a set of rules that help in manipulating logarithmic terms in an expression.
They are fundamental in mathematics to simplify and solve equations or expressions involving logarithms.Some of the key logarithmic properties include:
  • Power Rule: Allows transformation of coefficients to exponents (e.g., \(a \ln b = \ln(b^a)\)).
  • Addition Rule: Combines logs of products (e.g., \(\ln(a) + \ln(b) = \ln(a \times b)\)).
  • Subtraction Rule: This rule isn't part of this specific problem, but it's worth noting: \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\).
Using these properties, you can transform and condense logarithmic expressions for easier handling.
Mastering these properties is essential for dealing with complex logarithmic equations.

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Most popular questions from this chapter

Write an exponential equation that has a solution of \(x=4\). Then write a logarithmic equation that has a solution of \(x=-3\).

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