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Magazine Chapter 6: Problem 35
Since \(f(x)=e^{x}\) is a strictly increasing function, if \(a
Short Answer
Expert verified
The solution is \( x < 9.54275 \).
Step by step solution
01
Understanding the Inequality
The given inequality is \( \ln(2x+1) < 3 \). Our goal is to find the values of \( x \) that satisfy this inequality. Since \( \ln(x) \) is the natural logarithm, we begin by recalling that the exponential function \( e^{x} \) and the natural logarithm function \( \ln(x) \) are inverses of each other.
02
Applying the Exponential Function
To eliminate the \( \ln \) from the inequality \( \ln(2x+1) < 3 \), we exponentiate both sides using base \( e \). So, we raise \( e \) to each side of the inequality: \( e^{\ln(2x+1)} < e^{3} \). This simplifies to \( 2x+1 < e^{3} \) because \( e^{\ln(y)} = y \) for any positive \( y \).
03
Solving for x
Now, isolate \( x \) in the inequality \( 2x+1 < e^{3} \). Start by subtracting 1 from both sides: \( 2x < e^{3} - 1 \). Next, divide both sides by 2 to solve for \( x \): \( x < \frac{e^{3} - 1}{2} \).
04
Calculating \( e^{3} \)
To find an approximate solution, compute \( e^{3} \). \( e^{3} \approx 20.0855 \). Substitute this value in to get \( x < \frac{20.0855 - 1}{2} \), which simplifies to \( x < 9.54275 \).
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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, often represented as \( \ln(x) \), is a powerful mathematical function. It tells us the time it takes for a certain amount of continuous growth, such as compound interest or population growth, to reach a specific level. The natural logarithm is particularly special because it is the inverse of the exponential function with the base \( e \).
- The number \( e \) is an irrational constant approximately equal to 2.71828.
- In simpler terms, \( \ln(x) \) answers the question: "To what power must \( e \) be raised, to produce \( x \)?"
Exponential Function
The exponential function, represented as \( e^{x} \), is crucial in mathematics and various applications like finance, biology, and physics. This function grows at a rate proportional to its current value, which makes it fascinating and useful.
- In an inequality or equation, \( e^{x} \) can often transform challenging logarithmic expressions into something simpler.
- The exponential function inherently undoes the effect of a natural logarithm, so you can clear a logarithmic inequality by applying \( e^{(...)} \) on both sides.
Solving Inequalities
Solving inequalities is similar to solving equations, but with a key difference: the sign matters. When you solve the inequality \( \ln(2x+1) < 3 \), you're asking, "For which values of \( x \) is this inequality true?"
- Start by isolating the variable from the logarithmic expression. Use the exponential function to cancel the \( \ln \).
- After converting using the exponential function, you get \( 2x+1 < e^{3} \).
- From here, it's straightforward: subtract 1 and divide by 2 to isolate \( x \), solving \( x < \frac{e^{3} - 1}{2} \).
Inverse Functions
Inverse functions are a fascinating aspect of mathematics. They enable you to reverse a calculation that has been done. For instance, the natural logarithm \( \ln(x) \) and exponential function \( e^{x} \) are inverses, meaning applying one follows undoing the effect of the other:
- When \( y = \ln(x) \), we can find \( x \) by applying \( e^{y} \), such that \( x = e^{y} \).
- Similarly, if \( y = e^{x} \), then applying \( \ln(y) \) returns \( x \), the exponent.
