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Chapter 4: Problem 18

Simplify the following expressions. \(e^{\ln 3-2 \ln x}\)

Short Answer

Expert verified
\( \dfrac{3}{x^2} \).

Step by step solution

01

- Use Logarithm Properties

Recall the logarithmic property that states \(a \ln b = \ln (b^a)\). Using this, convert \(2 \ln x\) to \(\ln (x^2)\). The expression now is \( \ln 3 - \ln (x^2)\).
02

- Apply Logarithm Difference Rule

Utilize the logarithm rule \( \ln a - \ln b = \ln \left(\dfrac{a}{b}\right) \). This transforms \( \ln 3 - \ln (x^2)\) into \( \ln \( \dfrac{3}{x^2} \)\).
03

- Simplify the Exponential Expression

Now rewrite the expression \( e^{ \ln \( \dfrac{3}{x^2} \)} \) using the property that \( e^{ \ln y} = y \). Thus, \( e^{ \ln \( \dfrac{3}{x^2} \)} \) simplifies to \( \dfrac{3}{x^2} \).

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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 rules that help us manipulate logarithms more easily. One essential property is the power rule. It states that if you have a logarithm multiplied by a constant, you can move that constant as an exponent of the logarithmic argument:
  • \( a \, \ln b = \ln (b^a) \)
This property is extremely helpful in simplifying complex logarithmic expressions. For instance, in the exercise above, the term \(2 \ln x\) was converted to \(\ln (x^2) \) using this property, making it easier to work with the following operations.

Understanding and applying logarithm properties correctly can greatly simplify your calculations involving logarithms.
logarithm difference rule
The logarithm difference rule is another important property of logarithms. This rule states that the logarithm of a quotient is equal to the difference of the logarithms:
  • \( \ln a - \ln b = \ln \left( \, \dfrac{a}{b} \, \right) \)

    • This rule simplifies expressions where two logarithms with the same base are subtracted. In our exercise, after applying the power rule, we had the expression \( \ln 3 - \ln (x^2) \). Using the difference rule, this was transformed into a single logarithm: \( \ln \left( \, \dfrac{3}{x^2} \, \right) \).

    The logarithm difference rule helps to condense multiple logarithmic terms into a simpler form, making further simplifications more straightforward.
exponential expressions
An exponential expression involves variables in the exponent. A key property of exponential functions and natural logarithms (\(e\) and \(\ln\)) states that \( e^{ \ln y} = y \). This property is useful in reversing the natural logarithm when it is contained within an exponent.
  • For example, if you have \(e^{ \ln(a)}\), it simplifies directly to \(a\).
In our exercise, once the logarithmic difference rule was applied, we obtained the expression \(e^{ \ln \left( \, \dfrac{3}{x^2} \, \right) }\). Using the key property, this expression was simplified to \( \dfrac{3}{x^2} \).
This simplification is crucial for solving problems that intertwine logarithmic and exponential operations, making calculations more manageable.

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