Algebra Examples

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

Step 1

Write in exponential form.

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Step 1.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = e$ , $x = {(x + 1)}^{2}$ , and $y = 2$ .

$b = e$

$x = {(x + 1)}^{2}$

$y = 2$

Step 1.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$e^{2} = {(x + 1)}^{2}$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as ${(x + 1)}^{2} = e^{2}$ .

${(x + 1)}^{2} = e^{2}$

Step 2.2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| x + 1 | = | e |$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x + 1 = \pm | e |$

Step 2.3.2

$e$ is approximately $2.71828182$ which is positive so remove the absolute value

$x + 1 = \pm e$

Step 2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x + 1 = e$

Step 2.3.3.2

Subtract $1$ from both sides of the equation.

$x = e - 1$

Step 2.3.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + 1 = - e$

Step 2.3.3.4

Subtract $1$ from both sides of the equation.

$x = - e - 1$

Step 2.3.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = e - 1, - e - 1$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = e - 1, - e - 1$

Decimal Form:

$x = 1.71828182 \dots, - 3.71828182 \dots$