Problem 35 - Precalculus

Question

Since \(f(x)=e^{x}\) is a strictly increasing function, if \(a

Step-by-Step Solution

Answer

The solution is \( x < 9.54275 \).

Step 1: 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.

Step 2: 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 \( y \).

Step 3: 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 \( x \): \( x < \frac{e^{3} - 1}{2} \).

Step 4: 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 \( x < 9.54275 \).

Key Concepts

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 \( x \)?"

When you see \( \ln(x) \) in an inequality, you're often trying to find a range of values for \( x \) that satisfy the inequality, much like solving an equation. Understanding this relationship with \( e \) is key to manipulating and solving expressions involving natural logarithms.

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.

For example, in the inequality \( \ln(2x+1) < 3 \), applying \( e^{(...)}, \) converts this to \( 2x+1 < e^{3} \), removing the logarithmic complexity.

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 \( x < \frac{e^{3} - 1}{2} \).

Remember, always maintain focus on the inequality sign during your calculations. The nature of exponential and logarithmic functions helps you ensure that the direction doesn't flip unexpectedly.

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.

This interplay between logarithms and exponentials is widely used in solving inequalities. By understanding these inverse relationships, you can transform complex mathematical problems into more manageable forms and simplify expressions efficiently.