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

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

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Determine the direction of a vector scalar multiple

Full solution

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 $x \approx 4.69453$ .

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1

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2

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1

2

3

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2

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2

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10

6

10

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10

8

10

9

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1

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