Calculus Examples

$\ln (y - 1) - \ln (2) = x + \ln (x)$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{y - 1}{2}) = x + \ln (x)$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$\ln (\frac{y - 1}{2}) - \ln (x) = x$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{\frac{y - 1}{2}}{x}) = x$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{y - 1}{2} \cdot \frac{1}{x}) = x$

Step 5

Multiply $\frac{y - 1}{2}$ by $\frac{1}{x}$ .

$\ln (\frac{y - 1}{2 x}) = x$

Step 6

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{y - 1}{2 x})} = e^{x}$

Step 7

Rewrite $\ln (\frac{y - 1}{2 x}) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{x} = \frac{y - 1}{2 x}$

Step 8

Solve for $y$ .

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Step 8.1

Rewrite the equation as $\frac{y - 1}{2 x} = e^{x}$ .

$\frac{y - 1}{2 x} = e^{x}$

Step 8.2

Multiply both sides by $2 x$ .

$\frac{y - 1}{2 x} (2 x) = e^{x} (2 x)$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Simplify $\frac{y - 1}{2 x} (2 x)$ .

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Step 8.3.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{y - 1}{2 x} x = e^{x} (2 x)$

Step 8.3.1.1.2

Cancel the common factor of $2$ .

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Step 8.3.1.1.2.1

Factor $2$ out of $2 x$ .

$2 \frac{y - 1}{2 (x)} x = e^{x} (2 x)$

Step 8.3.1.1.2.2

Cancel the common factor.

$2 \frac{y - 1}{2 x} x = e^{x} (2 x)$

Step 8.3.1.1.2.3

Rewrite the expression.

$\frac{y - 1}{x} x = e^{x} (2 x)$

Step 8.3.1.1.3

Cancel the common factor of $x$ .

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Step 8.3.1.1.3.1

Cancel the common factor.

$\frac{y - 1}{x} x = e^{x} (2 x)$

Step 8.3.1.1.3.2

Rewrite the expression.

$y - 1 = e^{x} (2 x)$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $e^{x} (2 x)$ .

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Step 8.3.2.1.1

Rewrite using the commutative property of multiplication.

$y - 1 = 2 e^{x} x$

Step 8.3.2.1.2

Reorder factors in $2 e^{x} x$ .

$y - 1 = 2 x e^{x}$

Step 8.4

Add $1$ to both sides of the equation.

$y = 2 x e^{x} + 1$