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Chapter 4: Problem 35

Solve the following equations for \(x.\) $$\left(e^{2}\right)^{x} \cdot e^{\ln 1}=4$$

Short Answer

Expert verified
x = \(\frac{\ln(4)}{2}\)

Step by step solution

01

- Simplify the equation

Rewrite the equation to a simpler form. Note that \( e^{\ln 1} = 1 \) since \( \ln 1 = 0 \) and \( e^0 = 1 \). The equation simplifies to: \[ (e^2)^x \times 1 = 4 \] or \[ (e^2)^x = 4 \]
02

- Use properties of exponents

Convert \( (e^2)^x \) to \( e^{2x} \). The equation can now be written as: \[ e^{2x} = 4 \]
03

- Take natural logarithm on both sides

Apply the natural logarithm to both sides to solve for x: \[ \ln(e^{2x}) = \ln(4) \]
04

- Simplify using properties of logarithms

Use the property \( \ln(e^a) = a \). Thus, the equation simplifies to: \[ 2x = \ln(4) \]
05

- Solve for x

Isolate x by dividing both sides by 2: \[ x = \frac{\ln(4)}{2} \]

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Key Concepts

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

Natural Logarithm
The natural logarithm, denoted as \(\text{ln}\), is a logarithm with base e, where \(e \approx 2.71828\). It's also called the Napierian logarithm. It has special properties that make it particularly useful for solving exponential equations. For instance:
  • \(\text{ln}(e^a) = a\): This states that the natural logarithm of \(e\) raised to any power is simply the power itself.
  • \(\text{ln}(1) = 0\): This shows that the natural logarithm of 1 is always 0.
These properties simplify the process of dealing with exponential functions. In our exercise, recognizing that \( e^{\text{ln}1} = 1 \) helped with simplifying the equation right from the start, since \( \text{ln}1 = 0\) and therefore \( e^0 = 1 \). This step reduced our problem into a much simpler form, making the subsequent steps easier to follow.
Properties of Exponents
Understanding the properties of exponents is crucial when solving exponential equations like the one in our exercise. Some key properties include:
  • \(a^{m \cdot n} = (a^m)^n\): This allows us to rewrite an expression in a simpler form.
  • \(a^m \cdot a^n = a^{m+n}\): This is useful when combining terms with the same base.
In our problem, we used \( (e^2)^x = e^{2x} \) to transform the left side of the equation to a simpler exponential form. This step enabled us to apply the natural logarithm effectively in the next steps. Without these properties, manipulating and simplifying exponentials would be far more challenging.
Solving Equations
Solving equations, especially those involving exponential and logarithmic functions, often requires a methodical approach. Let's break down some general steps:
  • Simplify the Equation: Reduce the equation to its simplest form. In our example, we recognized \( e^{\text{ln}1} = 1 \) to simplify it initially.
  • Use Key Properties: Apply properties of logarithms and exponents to rearrange and further simplify the equation, as seen when we converted \( (e^2)^x \) to \( e^{2x} \).
  • Apply Logarithms: Use logarithms to linearize exponential equations. In our sequence, we utilized \( \text{ln}(e^{2x}) = 2x \) to solve for \ x \.
  • Isolate the Variable: Finally, solve for the variable by isolating it, as we did when we divided both sides by 2 to find \( x = \frac{\text{ln}(4)}{2} \).
By following these steps, you can methodically solve complex equations with greater ease. Contextual understanding of each step ensures you don't miss critical manipulations or rules.

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Most popular questions from this chapter

Differentiate. $$y=\ln \frac{x+1}{x-1}$$

Differentiate the following functions. $$y=8 e^{x}\left(1+2 e^{x}\right)^{2}$$

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

Find the values of \(x\) at which the function has a possible relative maximum or minimum point. (Recall that \(e^{x}\) is positive for all \(x .\) ) Use the second derivative to determine the nature of the function at these points. $$f(x)=\frac{3-4 x}{e^{2 x}}$$

Differentiate. $$y=\ln [\sqrt{x e^{x^{2}+1}}]$$
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