giving your answer correct to $3$ significant figures. $\newline$ $10$ . Solve the equation $\newline$ $\ln \left(1+x^{2}\right)=1+2 \ln x$ $\newline$ giving your answer correct to $3$ significant figures.

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Q. giving your answer correct to

3

significant figures.

\newline

10

. Solve the equation

\newline

\ln \left(1+x^{2}\right)=1+2 \ln x

\newline

giving your answer correct to

3

significant figures.

Combine logarithms: First, let's use the property of logarithms that says $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ to combine the right side of the equation. $\newline$ So we rewrite the equation as $\ln(1+x^{2}) = \ln(e) + \ln(x^2)$ .
Simplify equation: Now, since $\ln(e)$ is $1$ , we can simplify the equation to $\ln(1+x^{2}) = \ln(x^2) + 1$ .
Set up quadratic equation: Next, we use another property of logarithms that says if $\ln(a) = \ln(b)$ , then $a = b$ . So we set $1+x^{2}$ equal to $e\cdot(x^2)$ . $\newline$ This gives us the equation $1+x^{2} = e\cdot x^2$ .
Solve for x: Now we need to solve for x. Let's expand $e*x^2$ to $ex^2$ and move all terms to one side to get a quadratic equation. $\newline$ So we have $x^2 - ex^2 + 1 = 0$ .
Substitute e value: This simplifies to $(1-e)x^2 + 1 = 0$ .
Calculate $x$ : Now, we can subtract $1$ from both sides to get $(1-e)x^2 = -1$ .
Identify mistake: Next, we divide both sides by $(1-e)$ to solve for $x^2$ . So, $x^2 = -\frac{1}{(1-e)}$ .
Identify mistake: Next, we divide both sides by $(1-e)$ to solve for $x^2$ . So, $x^2 = -\frac{1}{(1-e)}$ . Since $e$ is approximately $2.718$ , we substitute that in to get $x^2 = -\frac{1}{(1-2.718)}$ .
Identify mistake: Next, we divide both sides by $(1-e)$ to solve for $x^2$ . So, $x^2 = -\frac{1}{(1-e)}$ . Since $e$ is approximately $2.718$ , we substitute that in to get $x^2 = -\frac{1}{(1-2.718)}$ . Calculating the denominator, we get $1 - 2.718 = -1.718$ .
Identify mistake: Next, we divide both sides by $(1-e)$ to solve for $x^2$ . So, $x^2 = -1/(1-e)$ . Since $e$ is approximately $2.718$ , we substitute that in to get $x^2 = -1/(1-2.718)$ . Calculating the denominator, we get $1 - 2.718 = -1.718$ . Now, we find $x^2 = -1/-1.718$ which simplifies to $x^2 = 1/1.718$ .
Identify mistake: Next, we divide both sides by $(1-e)$ to solve for $x^2$ . So, $x^2 = -\frac{1}{(1-e)}$ . Since $e$ is approximately $2.718$ , we substitute that in to get $x^2 = -\frac{1}{(1-2.718)}$ . Calculating the denominator, we get $1 - 2.718 = -1.718$ . Now, we find $x^2 = -\frac{1}{-1.718}$ which simplifies to $x^2 = \frac{1}{1.718}$ . Taking the square root of both sides to solve for $x$ , we get $x^2$ $0$ .
Identify mistake: Next, we divide both sides by $(1-e)$ to solve for $x^2$ . So, $x^2 = -\frac{1}{(1-e)}$ . Since $e$ is approximately $2.718$ , we substitute that in to get $x^2 = -\frac{1}{(1-2.718)}$ . Calculating the denominator, we get $1 - 2.718 = -1.718$ . Now, we find $x^2 = -\frac{1}{-1.718}$ which simplifies to $x^2 = \frac{1}{1.718}$ . Taking the square root of both sides to solve for $x$ , we get $x^2$ $0$ . Calculating the square root, we find $x^2$ $1$ .
Identify mistake: Next, we divide both sides by $(1-e)$ to solve for $x^2$ . So, $x^2 = -\frac{1}{(1-e)}$ . Since $e$ is approximately $2.718$ , we substitute that in to get $x^2 = -\frac{1}{(1-2.718)}$ . Calculating the denominator, we get $1 - 2.718 = -1.718$ . Now, we find $x^2 = -\frac{1}{-1.718}$ which simplifies to $x^2 = \frac{1}{1.718}$ . Taking the square root of both sides to solve for $x$ , we get $x^2$ $0$ . Calculating the square root, we find $x^2$ $1$ . However, we made a mistake. We cannot take the square root of a negative number when we are looking for real solutions. The correct step should have been to recognize that $x^2$ cannot be negative, so there must have been an error in our calculations. Let's go back and check our work.