Basic Math Examples

$\log (\frac{m^{2}}{n^{5}}) = 21 \log (m) - 5 \log (n)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $21 \log (m) - 5 \log (n)$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Simplify $21 \log (m)$ by moving $21$ inside the logarithm.

$\log (\frac{m^{2}}{n^{5}}) = \log (m^{21}) - 5 \log (n)$

Step 1.1.1.2

Simplify $- 5 \log (n)$ by moving $5$ inside the logarithm.

$\log (\frac{m^{2}}{n^{5}}) = \log (m^{21}) - \log (n^{5})$

Step 1.1.2

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

$\log (\frac{m^{2}}{n^{5}}) = \log (\frac{m^{21}}{n^{5}})$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$\frac{m^{2}}{n^{5}} = \frac{m^{21}}{n^{5}}$

Step 3

Solve for $m$ .

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Step 3.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$m^{2} = m^{21}$

Step 3.2

Subtract $m^{21}$ from both sides of the equation.

$m^{2} - m^{21} = 0$

Step 3.3

Factor $m^{2}$ out of $m^{2} - m^{21}$ .

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Step 3.3.1

Multiply by $1$ .

$m^{2} \cdot 1 - m^{21} = 0$

Step 3.3.2

Factor $m^{2}$ out of $- m^{21}$ .

$m^{2} \cdot 1 + m^{2} (- m^{19}) = 0$

Step 3.3.3

Factor $m^{2}$ out of $m^{2} \cdot 1 + m^{2} (- m^{19})$ .

$m^{2} (1 - m^{19}) = 0$

Step 3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$m^{2} = 0$

$1 - m^{19} = 0$

Step 3.5

Set $m^{2}$ equal to $0$ and solve for $m$ .

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Step 3.5.1

Set $m^{2}$ equal to $0$ .

$m^{2} = 0$

Step 3.5.2

Solve $m^{2} = 0$ for $m$ .

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Step 3.5.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$m = \pm \sqrt{0}$

Step 3.5.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.5.2.2.1

Rewrite $0$ as $0^{2}$ .

$m = \pm \sqrt{0^{2}}$

Step 3.5.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$m = \pm 0$

Step 3.5.2.2.3

Plus or minus $0$ is $0$ .

$m = 0$

Step 3.6

Set $1 - m^{19}$ equal to $0$ and solve for $m$ .

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Step 3.6.1

Set $1 - m^{19}$ equal to $0$ .

$1 - m^{19} = 0$

Step 3.6.2

Solve $1 - m^{19} = 0$ for $m$ .

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Step 3.6.2.1

Subtract $1$ from both sides of the equation.

$- m^{19} = - 1$

Step 3.6.2.2

Divide each term in $- m^{19} = - 1$ by $- 1$ and simplify.

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Step 3.6.2.2.1

Divide each term in $- m^{19} = - 1$ by $- 1$ .

$\frac{- m^{19}}{- 1} = \frac{- 1}{- 1}$

Step 3.6.2.2.2

Simplify the left side.

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Step 3.6.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{m^{19}}{1} = \frac{- 1}{- 1}$

Step 3.6.2.2.2.2

Divide $m^{19}$ by $1$ .

$m^{19} = \frac{- 1}{- 1}$

Step 3.6.2.2.3

Simplify the right side.

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Step 3.6.2.2.3.1

Divide $- 1$ by $- 1$ .

$m^{19} = 1$

Step 3.6.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$m = \sqrt[19]{1}$

Step 3.6.2.4

Any root of $1$ is $1$ .

$m = 1$

Step 3.7

The final solution is all the values that make $m^{2} (1 - m^{19}) = 0$ true.

$m = 0, 1$