Linear Algebra Examples

$5 x^{2 y} = 30$

Step 1

Divide each term in $5 x^{2 y} = 30$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 x^{2 y} = 30$ by $5$ .

$\frac{5 x^{2 y}}{5} = \frac{30}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 x^{2 y}}{5} = \frac{30}{5}$

Step 1.2.1.2

Divide $x^{2 y}$ by $1$ .

$x^{2 y} = \frac{30}{5}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $30$ by $5$ .

$x^{2 y} = 6$

Step 2

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

$\ln (x^{2 y}) = \ln (6)$

Step 3

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

$2 y \ln (x) = \ln (6)$

Step 4

Divide each term in $2 y \ln (x) = \ln (6)$ by $2 \ln (x)$ and simplify.

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Step 4.1

Divide each term in $2 y \ln (x) = \ln (6)$ by $2 \ln (x)$ .

$\frac{2 y \ln (x)}{2 \ln (x)} = \frac{\ln (6)}{2 \ln (x)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 y \ln (x)}{2 \ln (x)} = \frac{\ln (6)}{2 \ln (x)}$

Step 4.2.1.2

Rewrite the expression.

$\frac{y \ln (x)}{\ln (x)} = \frac{\ln (6)}{2 \ln (x)}$

Step 4.2.2

Cancel the common factor of $\ln (x)$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{y \ln (x)}{\ln (x)} = \frac{\ln (6)}{2 \ln (x)}$

Step 4.2.2.2

Divide $y$ by $1$ .

$y = \frac{\ln (6)}{2 \ln (x)}$

Step 5

Set the argument in $\ln (x)$ greater than $0$ to find where the expression is defined.

$x > 0$

Step 6

Set the denominator in $\frac{\ln (6)}{2 \ln (x)}$ equal to $0$ to find where the expression is undefined.

$2 \ln (x) = 0$

Step 7

Solve for $x$ .

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Step 7.1

Divide each term in $2 \ln (x) = 0$ by $2$ and simplify.

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Step 7.1.1

Divide each term in $2 \ln (x) = 0$ by $2$ .

$\frac{2 \ln (x)}{2} = \frac{0}{2}$

Step 7.1.2

Simplify the left side.

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Step 7.1.2.1

Cancel the common factor of $2$ .

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Step 7.1.2.1.1

Cancel the common factor.

$\frac{2 \ln (x)}{2} = \frac{0}{2}$

Step 7.1.2.1.2

Divide $\ln (x)$ by $1$ .

$\ln (x) = \frac{0}{2}$

Step 7.1.3

Simplify the right side.

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Step 7.1.3.1

Divide $0$ by $2$ .

$\ln (x) = 0$

Step 7.2

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

$e^{\ln (x)} = e^{0}$

Step 7.3

Rewrite $\ln (x) = 0$ 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 $b^{y} = x$ .

$e^{0} = x$

Step 7.4

Solve for $x$ .

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Step 7.4.1

Rewrite the equation as $x = e^{0}$ .

$x = e^{0}$

Step 7.4.2

Anything raised to $0$ is $1$ .

$x = 1$

Step 8

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(0, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x > 0, x \neq 1}$

Step 9