Problem 19 - Precalculus

Question

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$7 e^{-3 \ln x}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{7}{x^3}\).

Step 1: Understand the Expression

We begin with the expression \(7 e^{-3 \ln x}\). This expression involves the natural logarithm and an exponential function with a constant multiplier.

Step 2: Use the Power Rule of Logarithms

The power rule states that \(-3 \ln x = \ln(x^{-3})\). This converts the logarithmic expression into a simple exponent.

Step 3: Substitute the Logarithmic Identity into the Exponent

Substitute \(\ln(x^{-3})\) into the exponential, yielding \(e^{-3 \ln x} = e^{\ln (x^{-3})}\).

Step 4: Simplify Using the Inverse Properties of Exponentials and Logarithms

Utilize the fact that \(e^{\ln a} = a\) to further simplify \(e^{\ln(x^{-3})} = x^{-3}\). This works because the exponential and the logarithm are inverse functions.

Step 5: Final Simplification

Substitute back into the expression to get \(7 \cdot x^{-3}\), which is equivalent to \(\frac{7}{x^3}\). This is the simplified form of the expression.

Key Concepts

Power Rule

The power rule for logarithms is an essential tool for simplifying expressions involving exponents and logarithms. The rule states that if you have an exponent inside a logarithm, such as \(b \ln a\), it can be rewritten as \(\ln(a^b)\). This simple yet powerful rule helps in transforming a logarithmic expression into an expression with a straightforward exponent.

For example, if we have \(-3 \ln x\), applying the power rule gives \(\ln(x^{-3})\). This transformation is especially useful in simplifying expressions because it converts a more complex logarithmic form into a manageable exponential form.

• It allows for easier manipulation of expressions involving both logs and exponents.
• This rule is a foundation for further simplification using other logarithmic properties.

Understanding and applying the power rule is key when working with logarithms and exponentials. With practice, recognizing when to use this rule becomes second nature, simplifying many mathematical expressions.

Exponential Function

Exponential functions are mathematical functions of the form \(f(x) = a^x\), where \(a\) is a constant and \(x\) is a variable. In the context of logarithms, the base \(e\) (approximately 2.718) becomes particularly important, known as the natural exponential function \(e^x\).

This function is unique due to its rate of growth, which is directly proportional to its current value, making it widely applicable in real-world scenarios, from population growth to compound interest calculations. When combined with the natural logarithm \(\ln(x)\), it forms a vital part of calculus and other advanced math fields.

• In expressions like \(e^{-3 \ln x}\), you encounter both the exponential function and logarithmic properties.
• The exponential function acts as a bridge to convert logarithmic expressions back to their original base form.

Mastering exponential functions, especially those involving \(e\), is crucial for solving and simplifying complex logarithmic expressions. Recognizing these patterns helps in working through many mathematical problems efficiently.

Inverse Properties

Understanding the inverse properties of exponentials and logarithms is essential when simplifying expressions like \(e^{\ln a}\). The key property here is that the exponential function \(e^x\) and the natural logarithm \(\ln x\) are inverses. This means applying one after the other "undoes" the operation.

In practical terms, the relationship \(e^{\ln a} = a\) is used to simplify expressions. For instance, turning \(e^{\ln(x^{-3})}\) into \(x^{-3}\) by recognizing \(\ln (x^{-3})\) as exactly what's needed to cancel the exponential operation.

• This property provides a direct way to simplify "nested" logarithmic-exponential expressions.
• It's the backbone of moving between logarithmic and exponential forms.

These inverse properties are particularly helpful for dealing with complex equations. They enable straightforward simplification steps, making seemingly difficult problems more manageable.