Calculus Examples

$4 x y + \ln (x^{2} y) = 2$

Step 1

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

$e^{\ln (x^{2} y)} = e^{2 - 4 x y}$

Step 2

Rewrite $\ln (x^{2} y) = 2 - 4 x y$ 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 $b^{y} = x$ .

$e^{2 - 4 x y} = x^{2} y$

Step 3

Solve for $x$ .

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Step 3.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{2 - 4 x y}) = \ln (x^{2} y)$

Step 3.2

Expand the left side.

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Step 3.2.1

Expand $\ln (e^{2 - 4 x y})$ by moving $2 - 4 x y$ outside the logarithm.

$(2 - 4 x y) \ln (e) = \ln (x^{2} y)$

Step 3.2.2

The natural logarithm of $e$ is $1$ .

$(2 - 4 x y) \cdot 1 = \ln (x^{2} y)$

Step 3.2.3

Multiply $2 - 4 x y$ by $1$ .

$2 - 4 x y = \ln (x^{2} y)$

Step 3.3

Subtract $\ln (x^{2} y)$ from both sides of the equation.

$2 - 4 x y - \ln (x^{2} y) = 0$

Step 3.4

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

$e^{\ln (x^{2} y)} = e^{2 - 4 x y}$

Step 3.5

$e^{2 - 4 x y} = x^{2} y$

Step 3.6

Solve for $x$ .

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Step 3.6.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{2 - 4 x y}) = \ln (x^{2} y)$

Step 3.6.2

Expand the left side.

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Step 3.6.2.1

Expand $\ln (e^{2 - 4 x y})$ by moving $2 - 4 x y$ outside the logarithm.

$(2 - 4 x y) \ln (e) = \ln (x^{2} y)$

Step 3.6.2.2

The natural logarithm of $e$ is $1$ .

$(2 - 4 x y) \cdot 1 = \ln (x^{2} y)$

Step 3.6.2.3

Multiply $2 - 4 x y$ by $1$ .

$2 - 4 x y = \ln (x^{2} y)$

Step 3.6.3

Subtract $\ln (x^{2} y)$ from both sides of the equation.

$2 - 4 x y - \ln (x^{2} y) = 0$

Step 3.6.4

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

$e^{\ln (x^{2} y)} = e^{2 - 4 x y}$

Step 3.6.5

$e^{2 - 4 x y} = x^{2} y$

Step 3.6.6

Solve for $x$ .

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Step 3.6.6.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{2 - 4 x y}) = \ln (x^{2} y)$

Step 3.6.6.2

Expand the left side.

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Step 3.6.6.2.1

Expand $\ln (e^{2 - 4 x y})$ by moving $2 - 4 x y$ outside the logarithm.

$(2 - 4 x y) \ln (e) = \ln (x^{2} y)$

Step 3.6.6.2.2

The natural logarithm of $e$ is $1$ .

$(2 - 4 x y) \cdot 1 = \ln (x^{2} y)$

Step 3.6.6.2.3

Multiply $2 - 4 x y$ by $1$ .

$2 - 4 x y = \ln (x^{2} y)$

Step 3.6.6.3

Subtract $\ln (x^{2} y)$ from both sides of the equation.

$2 - 4 x y - \ln (x^{2} y) = 0$

Step 3.6.6.4

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

$e^{\ln (x^{2} y)} = e^{2 - 4 x y}$

Step 3.6.6.5

$e^{2 - 4 x y} = x^{2} y$

Step 3.6.6.6

Solve for $x$ .

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Step 3.6.6.6.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{2 - 4 x y}) = \ln (x^{2} y)$

Step 3.6.6.6.2

Expand the left side.

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Step 3.6.6.6.2.1

Expand $\ln (e^{2 - 4 x y})$ by moving $2 - 4 x y$ outside the logarithm.

$(2 - 4 x y) \ln (e) = \ln (x^{2} y)$

Step 3.6.6.6.2.2

The natural logarithm of $e$ is $1$ .

$(2 - 4 x y) \cdot 1 = \ln (x^{2} y)$

Step 3.6.6.6.2.3

Multiply $2 - 4 x y$ by $1$ .

$2 - 4 x y = \ln (x^{2} y)$