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Chapter 4: Problem 104

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

Short Answer

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

Step by step solution

01

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 \( \ln (8x^{4} / 2x^{2}) \)
02

Simplify the denominator and the numerator

Simplify \( 8x^{4} / 2x^{2} \) to obtain \( 4x^{2} \)
03

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 \( 4x^{2} \).

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Key Concepts

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

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.

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Most popular questions from this chapter

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log (2 x+5)-\log x $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{9}(9 x) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \sqrt{e x} $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 2 \ln x-\frac{1}{2} \ln y $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b} x^{7} $$
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