Basic Math Examples

$m^{6 - 2 n} = 1$

Step 1

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

$\ln (m^{6 - 2 n}) = \ln (1)$

Step 2

Expand $\ln (m^{6 - 2 n})$ by moving $6 - 2 n$ outside the logarithm.

$(6 - 2 n) \ln (m) = \ln (1)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$6 \ln (m) - 2 n \ln (m) = \ln (1)$

Step 4

Simplify the right side.

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Step 4.1

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

$6 \ln (m) - 2 n \ln (m) = 0$

Step 5

Reorder $6 \ln (m)$ and $- 2 n \ln (m)$ .

$- 2 n \ln (m) + 6 \ln (m) = 0$

Step 6

Subtract $6 \ln (m)$ from both sides of the equation.

$- 2 n \ln (m) = - 6 \ln (m)$

Step 7

Divide each term in $- 2 n \ln (m) = - 6 \ln (m)$ by $- 2 \ln (m)$ and simplify.

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Step 7.1

Divide each term in $- 2 n \ln (m) = - 6 \ln (m)$ by $- 2 \ln (m)$ .

$\frac{- 2 n \ln (m)}{- 2 \ln (m)} = \frac{- 6 \ln (m)}{- 2 \ln (m)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 2 n \ln (m)}{- 2 \ln (m)} = \frac{- 6 \ln (m)}{- 2 \ln (m)}$

Step 7.2.1.2

Rewrite the expression.

$\frac{n \ln (m)}{\ln (m)} = \frac{- 6 \ln (m)}{- 2 \ln (m)}$

Step 7.2.2

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

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Step 7.2.2.1

Cancel the common factor.

$\frac{n \ln (m)}{\ln (m)} = \frac{- 6 \ln (m)}{- 2 \ln (m)}$

Step 7.2.2.2

Divide $n$ by $1$ .

$n = \frac{- 6 \ln (m)}{- 2 \ln (m)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $- 6$ and $- 2$ .

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Step 7.3.1.1

Factor $- 2$ out of $- 6 \ln (m)$ .

$n = \frac{- 2 (3 \ln (m))}{- 2 \ln (m)}$

Step 7.3.1.2

Cancel the common factors.

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Step 7.3.1.2.1

Factor $- 2$ out of $- 2 \ln (m)$ .

$n = \frac{- 2 (3 \ln (m))}{- 2 \ln (m)}$

Step 7.3.1.2.2

Cancel the common factor.

$n = \frac{- 2 (3 \ln (m))}{- 2 \ln (m)}$

Step 7.3.1.2.3

Rewrite the expression.

$n = \frac{3 \ln (m)}{\ln (m)}$

Step 7.3.2

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

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Step 7.3.2.1

Cancel the common factor.

$n = \frac{3 \ln (m)}{\ln (m)}$

Step 7.3.2.2

Divide $3$ by $1$ .

$n = 3$