Algebra Examples

${(\frac{x^{8}}{y z^{5}})}^{m} = 1$

Step 1

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

$\ln ({(\frac{x^{8}}{y z^{5}})}^{m}) = \ln (1)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({(\frac{x^{8}}{y z^{5}})}^{m})$ by moving $m$ outside the logarithm.

$m \ln (\frac{x^{8}}{y z^{5}}) = \ln (1)$

Step 2.2

Rewrite $\ln (\frac{x^{8}}{y z^{5}})$ as $\ln (x^{8}) - \ln (y z^{5})$ .

$m (\ln (x^{8}) - \ln (y z^{5})) = \ln (1)$

Step 2.3

Expand $\ln (x^{8})$ by moving $8$ outside the logarithm.

$m (8 \ln (x) - \ln (y z^{5})) = \ln (1)$

Step 2.4

Rewrite $\ln (y z^{5})$ as $\ln (y) + \ln (z^{5})$ .

$m (8 \ln (x) - (\ln (y) + \ln (z^{5}))) = \ln (1)$

Step 2.5

Expand $\ln (z^{5})$ by moving $5$ outside the logarithm.

$m (8 \ln (x) - (\ln (y) + 5 \ln (z))) = \ln (1)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $m (8 \ln (x) - (\ln (y) + 5 \ln (z)))$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$m (8 \ln (x) - \ln (y) - (5 \ln (z))) = \ln (1)$

Step 3.1.1.2

Multiply $5$ by $- 1$ .

$m (8 \ln (x) - \ln (y) - 5 \ln (z)) = \ln (1)$

Step 3.1.2

Apply the distributive property.

$m (8 \ln (x)) + m (- \ln (y)) + m (- 5 \ln (z)) = \ln (1)$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Rewrite using the commutative property of multiplication.

$8 m \ln (x) + m (- \ln (y)) + m (- 5 \ln (z)) = \ln (1)$

Step 3.1.3.2

Rewrite using the commutative property of multiplication.

$8 m \ln (x) - m \ln (y) + m (- 5 \ln (z)) = \ln (1)$

Step 3.1.3.3

Rewrite using the commutative property of multiplication.

$8 m \ln (x) - m \ln (y) - 5 m \ln (z) = \ln (1)$

Step 4

Simplify the right side.

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Step 4.1

The natural logarithm of $1$ is $0$ .

$8 m \ln (x) - m \ln (y) - 5 m \ln (z) = 0$

Step 5

Factor $m$ out of $8 m \ln (x) - m \ln (y) - 5 m \ln (z)$ .

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Step 5.1

Factor $m$ out of $8 m \ln (x)$ .

$m (8 \ln (x)) - m \ln (y) - 5 m \ln (z) = 0$

Step 5.2

Factor $m$ out of $- m \ln (y)$ .

$m (8 \ln (x)) + m (- 1 \ln (y)) - 5 m \ln (z) = 0$

Step 5.3

Factor $m$ out of $- 5 m \ln (z)$ .

$m (8 \ln (x)) + m (- 1 \ln (y)) + m (- 5 \ln (z)) = 0$

Step 5.4

Factor $m$ out of $m (8 \ln (x)) + m (- 1 \ln (y))$ .

$m (8 \ln (x) - 1 \ln (y)) + m (- 5 \ln (z)) = 0$

Step 5.5

Factor $m$ out of $m (8 \ln (x) - 1 \ln (y)) + m (- 5 \ln (z))$ .

$m (8 \ln (x) - 1 \ln (y) - 5 \ln (z)) = 0$

Step 6

Rewrite $- 1 \ln (y)$ as $- \ln (y)$ .

$m (8 \ln (x) - \ln (y) - 5 \ln (z)) = 0$

Step 7

Divide each term in $m (8 \ln (x) - \ln (y) - 5 \ln (z)) = 0$ by $8 \ln (x) - \ln (y) - 5 \ln (z)$ and simplify.

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Step 7.1

Divide each term in $m (8 \ln (x) - \ln (y) - 5 \ln (z)) = 0$ by $8 \ln (x) - \ln (y) - 5 \ln (z)$ .

$\frac{m (8 \ln (x) - \ln (y) - 5 \ln (z))}{8 \ln (x) - \ln (y) - 5 \ln (z)} = \frac{0}{8 \ln (x) - \ln (y) - 5 \ln (z)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $8 \ln (x) - \ln (y) - 5 \ln (z)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{m (8 \ln (x) - \ln (y) - 5 \ln (z))}{8 \ln (x) - \ln (y) - 5 \ln (z)} = \frac{0}{8 \ln (x) - \ln (y) - 5 \ln (z)}$

Step 7.2.1.2

Divide $m$ by $1$ .

$m = \frac{0}{8 \ln (x) - \ln (y) - 5 \ln (z)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $8 \ln (x) - \ln (y) - 5 \ln (z)$ .

$m = 0$