Problem 104 - College Algebra

Question

Write as a single term that does not contain a logarithm: $$ e^{\ln 8 x^{4}-\ln 2 x^{2}} $$

Step-by-Step Solution

Answer

The given expression simplifies to \( 4x^{2} \).

Step 1: Simplify the expression inside the power of \( e \)

Using the rule \( \ln a - \ln b = \ln(a/b) \), the power \( \ln 8 x^{4} - \ln 2 x^{2} \) can be rewritten as \( \ln (8x^{4} / 2x^{2}) \)

Step 2: Simplify the denominator and the numerator

Simplify \( 8x^{4} / 2x^{2} \) to obtain \( 4x^{2} \)

Step 3: Cancel out \( e^{} \) and \( \ln \)

With the power now as \( \ln 4x^{2} \), the expression itself simplifies to \( e^{\ln 4x^{2}} \). Using the property \( e^{ln a} = a \), we find the answer to be \( 4x^{2} \).

Key Concepts

Properties of Logarithms

Logarithms have specific properties that make them useful for simplifying complex expressions. Understanding these properties can greatly aid in the transformation and solving of logarithmic and exponential equations. Here are some key properties you should remember:

• Product Property: The logarithm of a product is the sum of the logarithms, expressed as \( \ln(ab) = \ln a + \ln b \).
• Quotient Property: The logarithm of a quotient is the difference of the logarithms, given by \( \ln(a/b) = \ln a - \ln b \). This property was crucial in simplifying our original problem.
• Power Property: The logarithm of a power can be rewritten by bringing the exponent in front: \( \ln(a^b) = b \ln a \).

These properties simplify complex logarithmic expressions, assisting in making calculations more manageable. For example, in the original exercise, we used the quotient property to combine the logarithms into a single logarithm term.

Simplifying Expressions

Simplifying expressions, particularly those with logarithms and exponents, is an essential algebraic skill. It involves rewriting expressions in a simpler or more understandable form. In the original exercise:

• We started with the expression \( \ln 8 x^{4}-\ln 2 x^{2} \).
• Using the quotient property \( \ln a - \ln b = \ln(a/b) \), it transformed into \( \ln \left( \frac{8x^{4}}{2x^{2}} \right) \).
• Upon simplifying the fraction \(\frac{8x^4}{2x^2} \), we ended up with \(4x^2\).

By simplifying the expression, complex terms are reduced, making the final expression more concise. It turns challenging equations into ones that are easier to understand and solve.

Exponential Functions

Exponential functions involve expressions where a constant base is raised to a variable exponent. In many mathematical problems, exponentials appear in combination with logarithms, as they are inverse functions of each other.

• The natural exponential function is often denoted by \(e\), and it has unique properties that simplify expressions when paired with natural logarithms \( \ln \).
• One crucial property is \( e^{\ln a} = a \), which allows you to "cancel" the \(e\) and \(\ln\), simplifying the expression within.
• This property was used in the final step of our original problem, turning \( e^{\ln 4x^2} \) into \( 4x^2 \).

Recognizing and applying this relationship between exponentials and logarithms can streamline the process of solving these types of mathematical problems, providing clarity and simplicity in results.