Problem 29 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 7 through 32, solve for \(x .\) $$ \left[\frac{3}{\ln (2 x+1)}\right]^{2}-1=10 $$

Step-by-Step Solution

Answer

The solution for the equation, i.e., the value of \(x\), is \(x = \frac{e^\frac{3}{\sqrt{11}} - 1}{2}\).

Step 1: Move 1 to right side of the equation

To start simply, move 1 from the left side to the right side to have \(\left[\frac{3}{\ln(2x+1)}\right]^2 = 10+1=11.\)

Step 2: Simplify and isolate logarithm

To get the logarithm by itself, take the square root of both sides to get \(\frac{3}{\ln(2x+1)} = \sqrt{11}\) and then, isolate the logarithm by taking the reciprocal of both sides: \(\ln(2x+1) = \frac{3}{\sqrt{11}}\).

Step 3: Using properties of logarithms

Applying exponentiation to both sides to neutralize the logarithm on the left side gets \(2x+1 = e^\frac{3}{\sqrt{11}}\). Algebraic transformations can be applied to solve for \(x\).

Step 4: Solve for \(x\)

Subtract 1 from both sides and divide by 2 to solve for \(x\). \(x = \frac{e^\frac{3}{\sqrt{11}} - 1}{2}\)

Key Concepts

Logarithmic equations

Logarithmic equations involve logarithmic expressions that are set equal to each other or to some other value, typically in a form like \( \log_b(x) = c \). These equations often require the use of exponential forms for solutions. Recognizing the structure of a logarithmic equation is crucial because it guides how to manipulate and eventually solve the equation. When solving these types of equations, the primary objective is to isolate the logarithmic part so that you can remove (or neutralize) the logarithmic function. For example:

• Transform \(\log_b(x) = c\) to an exponential form \(x = b^c\).
• Utilize exponentiation to both sides of the equation to eliminate the log, leading to an equation you can solve, such as \(2x+1 = e^{\frac{3}{\sqrt{11}}}\).

Approaching logarithmic equations requires patience and attention to detail in applying mathematical laws properly.

Algebraic transformations

Algebraic transformations are techniques used to rearrange and simplify equations. Applying these transformations can often help reveal the underlying simplicity of complex equations, making them easier to solve. In our problem, after isolating \(\ln(2x+1) = \frac{3}{\sqrt{11}}\), we performed an exponential transformation to simplify the equation.

• Transform \(e^{\ln(2x+1)}\) into \(2x+1\) using the fact that \(e\) raised to a \(\ln\) is the original value.
• Next, algebraically solve for \(x\) by "undoing" other operations, like subtracting or dividing, as shown when getting \(x = \frac{e^{\frac{3}{\sqrt{11}}} - 1}{2}\).

These transformations require a systematic approach to isolate variables, simplifying the path to finding solutions.

Properties of logarithms

Understanding the properties of logarithms is key in solving logarithmic equations and transforming them into a solvable form. Logarithms have several important properties that make them invaluable in calculus and algebra. Knowing these properties allows for effective manipulation of logarithmic expressions. Key properties include:

• Product Property: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Property: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)
• Power Property: \(\log_b(M^p) = p \cdot \log_b(M)\)

When solving an equation involving logarithms, these properties enable combining or breaking apart logarithmic expressions to simplify the solution process, such as reducing the equation to \(\ln(2x+1) = \frac{3}{\sqrt{11}}\) from which you can proceed further to solve the problem. Familiarity with these properties makes dealing with equations containing logs much more manageable.