Find the value of x that solves the equation ln(x-1)=2-ln 2. Answer:

Understand and Isolate Variable Term: Understand the equation and isolate the variable term. We have the equation ln(x-1) = 2 - ln 2. To solve for x, we need to isolate the term containing x on one side of the equation.Combine Right Side Terms: Use properties of logarithms to combine the terms on the right side of the equation. We can use the property of logarithms that states ln(a) - ln(b) = ln(a/b) to combine the terms on the right side of the equation. This gives us ln(x-1) = ln(e^2/2).Equate Arguments of Logarithms: Since the natural logarithm function ln is the inverse of the exponential function with base e, we can equate the arguments of the logarithms to solve for x. If ln(x-1) = ln(e^2/2), then x-1 = e^2/2.Solve for x: Solve for x by adding 1 to both sides of the equation. x = e^2/2 + 1Calculate Final Value: Calculate the value of e^2/2 + 1. We know that e is approximately 2.71828, so e^2 is approximately 7.38906. Dividing this by 2 gives us approximately 3.69453. Adding 1 to this gives us x ≈ 4.69453.

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Q. Find the value of

x

that solves the equation

\ln (x-1)=2-\ln 2

\newline

Answer:

Understand and Isolate Variable Term: Understand the equation and isolate the variable term. $\newline$ We have the equation $\ln(x-1) = 2 - \ln 2$ . To solve for $x$ , we need to isolate the term containing $x$ on one side of the equation.
Combine Right Side Terms: Use properties of logarithms to combine the terms on the right side of the equation. $\newline$ We can use the property of logarithms that states $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ to combine the terms on the right side of the equation. This gives us $\ln(x-1) = \ln\left(\frac{e^2}{2}\right)$ .
Equate Arguments of Logarithms: Since the natural logarithm function $\ln$ is the inverse of the exponential function with base $e$ , we can equate the arguments of the logarithms to solve for $x$ . If $\ln(x-1) = \ln(\frac{e^2}{2})$ , then $x-1 = \frac{e^2}{2}$ .
Solve for x: Solve for x by adding $1$ to both sides of the equation. $x = \frac{e^2}{2} + 1$
Calculate Final Value: Calculate the value of $e^2/2 + 1$ . We know that $e$ is approximately $2.71828$ , so $e^2$ is approximately $7.38906$ . Dividing this by $2$ gives us approximately $3.69453$ . Adding $1$ to this gives us $x \approx 4.69453$ .