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Chapter 4: Problem 14

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

Short Answer

Expert verified
Simplified expression is \(2^x\).

Step by step solution

01

Recall the properties of exponents and logarithms

Remember that for any base and exponents, if you have an expression in the form of \(e^{a \times b}\), it can be written as \((e^a)^b\).
02

Apply the property

Given the expression \(e^{x \times \text{ln} \thinspace 2}\), rewrite it using the exponent property: \(e^{x \times \text{ln} \thinspace 2} = (e^{\text{ln} \thinspace 2})^x\).
03

Simplify using the fact that \(e^{\text{ln} \thinspace k} = k\)

Since \(e^{\text{ln} \thinspace 2} = 2\), the expression simplifies to \(2^x\).

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Key Concepts

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

Properties of Exponents
Understanding the properties of exponents is crucial when simplifying exponential expressions. One of the key rules is that when you have an exponent raised to another exponent, like \( (a^m)^n \), it simplifies to \( a^{m \times n} \). This property helps break down complex expressions into more manageable parts.
  • For example, \( (e^a)^b = e^{a \times b} \).
  • Similarly, \( (2^3)^2 \) simplifies to \( 2^{3 \times 2} = 2^6 \).
Using these properties can make solving problems much simpler. Remember, if you see repeated multiplication inside an exponent, use the property to combine them effectively.
Logarithm Rules
Logarithms can initially seem complicated, but understanding a few key rules makes working with them easier. One important rule is that the natural log of a number (\text{ln}) has a direct relationship with exponentials. Specifically, \( e^{\text{ln} \thinspace a} = a \).
  • This rule is especially useful because it provides a straightforward way to simplify expressions that involve both exponents and logarithms.
  • For example, \( e^{\text{ln} \thinspace 2} = 2 \), which directly transforms an exponential-laden expression into a simpler form.
By applying this rule, you can easily connect exponents and logarithms, making seemingly complex problems much more manageable.
Simplification Techniques
Simplifying mathematical expressions often requires using a combination of properties and rules to break down complex forms into simpler ones. Consider the original problem: \( e^{x \text{ln} \thinspace 2} \).
  • First, recall the exponential property \( e^{a \times b} = (e^a)^b \).
  • Rewrite \( e^{x \times \text{ln} \thinspace 2} \) as \( (e^{\text{ln} \thinspace 2})^x \).
  • Next, use the fact that \( e^{\text{ln} \thinspace 2} = 2 \) to simplify this to \( 2^x \).
This step-by-step application of properties and rules ensures that each simplification is clear and correctly executed. Simplifying expressions becomes less daunting when approached methodically with these techniques.

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Most popular questions from this chapter

Differentiate. \(y=\ln \left[(1+x)^{2}(2+x)^{3}(3+x)^{4}\right]\)

Differentiate. \(y=\ln \left[e^{2 x}\left(x^{3}+1\right)\left(x^{4}+5 x\right)\right]\)

Graph the function \(f(x)=\ln |x|\) in the window \([-5,5]\) by \([-2,2]\), and verify that the derivative formula gives the correct values of \(f^{\prime}(x)\) at \(x=\pm 1, \pm 2\), and \(\pm 4\).

Differentiate the following functions. \(y=e^{\ln x+x}\)

Differentiate. \(y=\ln [(x+1)(2 x+1)(3 x+1)]\)
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