Basic Math Examples

$a^{q - r} \cdot b^{r - p} \cdot c^{p - q} = 1$

Step 1

Divide each term in $a^{q - r} \cdot b^{r - p} \cdot c^{p - q} = 1$ by $b^{r - p}$ and simplify.

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Step 1.1

Divide each term in $a^{q - r} \cdot b^{r - p} \cdot c^{p - q} = 1$ by $b^{r - p}$ .

$\frac{a^{q - r} \cdot b^{r - p} \cdot c^{p - q}}{b^{r - p}} = \frac{1}{b^{r - p}}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $b^{r - p}$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{a^{q - r} \cdot b^{r - p} \cdot c^{p - q}}{b^{r - p}} = \frac{1}{b^{r - p}}$

Step 1.2.1.2

Divide $a^{q - r} \cdot c^{p - q}$ by $1$ .

$a^{q - r} \cdot c^{p - q} = \frac{1}{b^{r - p}}$

$a^{q - r} c^{p - q} = \frac{1}{b^{r - p}}$

Step 2

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

$\ln (a^{q - r} c^{p - q}) = \ln (\frac{1}{b^{r - p}})$

Step 3

Expand the left side.

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Step 3.1

Rewrite $\ln (a^{q - r} c^{p - q})$ as $\ln (a^{q - r}) + \ln (c^{p - q})$ .

$\ln (a^{q - r}) + \ln (c^{p - q}) = \ln (\frac{1}{b^{r - p}})$

Step 3.2

Expand $\ln (a^{q - r})$ by moving $q - r$ outside the logarithm.

$(q - r) \ln (a) + \ln (c^{p - q}) = \ln (\frac{1}{b^{r - p}})$

Step 3.3

Expand $\ln (c^{p - q})$ by moving $p - q$ outside the logarithm.

$(q - r) \ln (a) + (p - q) \ln (c) = \ln (\frac{1}{b^{r - p}})$

Step 4

Expand the right side.

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Step 4.1

Rewrite $\ln (\frac{1}{b^{r - p}})$ as $\ln (1) - \ln (b^{r - p})$ .

$(q - r) \ln (a) + (p - q) \ln (c) = \ln (1) - \ln (b^{r - p})$

Step 4.2

Expand $\ln (b^{r - p})$ by moving $r - p$ outside the logarithm.

$(q - r) \ln (a) + (p - q) \ln (c) = \ln (1) - ((r - p) \ln (b))$

Step 4.3

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

$(q - r) \ln (a) + (p - q) \ln (c) = 0 - ((r - p) \ln (b))$

Step 4.4

Subtract $(r - p) \ln (b)$ from $0$ .

$(q - r) \ln (a) + (p - q) \ln (c) = - ((r - p) \ln (b))$

Step 4.5

Remove parentheses.

$(q - r) \ln (a) + (p - q) \ln (c) = - (r - p) \ln (b)$

Step 5

Simplify the left side.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

$q \ln (a) - r \ln (a) + (p - q) \ln (c) = - (r - p) \ln (b)$

Step 5.1.2

Apply the distributive property.

$q \ln (a) - r \ln (a) + p \ln (c) - q \ln (c) = - (r - p) \ln (b)$

Step 6

Simplify the right side.

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Step 6.1

Simplify $- (r - p) \ln (b)$ .

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Step 6.1.1

Apply the distributive property.

$q \ln (a) - r \ln (a) + p \ln (c) - q \ln (c) = (- r - - p) \ln (b)$

Step 6.1.2

Multiply $- - p$ .

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Step 6.1.2.1

Multiply $- 1$ by $- 1$ .

$q \ln (a) - r \ln (a) + p \ln (c) - q \ln (c) = (- r + 1 p) \ln (b)$

Step 6.1.2.2

Multiply $p$ by $1$ .

$q \ln (a) - r \ln (a) + p \ln (c) - q \ln (c) = (- r + p) \ln (b)$

Step 6.1.3

Apply the distributive property.

$q \ln (a) - r \ln (a) + p \ln (c) - q \ln (c) = - r \ln (b) + p \ln (b)$

Step 7

Simplify the left side.

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Step 7.1

Move $- r \ln (a)$ .

$q \ln (a) + p \ln (c) - q \ln (c) - r \ln (a) = - r \ln (b) + p \ln (b)$

Step 7.2

Reorder $q \ln (a)$ and $p \ln (c)$ .

$p \ln (c) + q \ln (a) - q \ln (c) - r \ln (a) = - r \ln (b) + p \ln (b)$

Step 8

Reorder $- r \ln (b)$ and $p \ln (b)$ .

$p \ln (c) + q \ln (a) - q \ln (c) - r \ln (a) = p \ln (b) - r \ln (b)$

Step 9

Move all the terms containing a logarithm to the left side of the equation.

$p \ln (c) + q \ln (a) - q \ln (c) - r \ln (a) - p \ln (b) + r \ln (b) = 0$

Step 10

Move all terms not containing $q$ to the right side of the equation.

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Step 10.1

Subtract $p \ln (c)$ from both sides of the equation.

$q \ln (a) - q \ln (c) - r \ln (a) - p \ln (b) + r \ln (b) = - p \ln (c)$

Step 10.2

Add $r \ln (a)$ to both sides of the equation.

$q \ln (a) - q \ln (c) - p \ln (b) + r \ln (b) = - p \ln (c) + r \ln (a)$

Step 10.3

Add $p \ln (b)$ to both sides of the equation.

$q \ln (a) - q \ln (c) + r \ln (b) = - p \ln (c) + r \ln (a) + p \ln (b)$

Step 10.4

Subtract $r \ln (b)$ from both sides of the equation.

$q \ln (a) - q \ln (c) = - p \ln (c) + r \ln (a) + p \ln (b) - r \ln (b)$

Step 11

Factor $q$ out of $q \ln (a) - q \ln (c)$ .

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Step 11.1

Factor $q$ out of $q \ln (a)$ .

$q (\ln (a)) - q \ln (c) = - p \ln (c) + r (\ln (a)) + p \ln (b) - r \ln (b)$

Step 11.2

Factor $q$ out of $- q \ln (c)$ .

$q (\ln (a)) + q (- 1 \ln (c)) = - p \ln (c) + r (\ln (a)) + p \ln (b) - r \ln (b)$

Step 11.3

Factor $q$ out of $q (\ln (a)) + q (- 1 \ln (c))$ .

$q (\ln (a) - 1 \ln (c)) = - p \ln (c) + r \ln (a) + p \ln (b) - r \ln (b)$

Step 12

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

$q (\ln (a) - \ln (c)) = - p \ln (c) + r \ln (a) + p \ln (b) - r \ln (b)$

Step 13

Divide each term in $q (\ln (a) - \ln (c)) = - p \ln (c) + r \ln (a) + p \ln (b) - r \ln (b)$ by $\ln (a) - \ln (c)$ and simplify.

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Step 13.1

Divide each term in $q (\ln (a) - \ln (c)) = - p \ln (c) + r \ln (a) + p \ln (b) - r \ln (b)$ by $\ln (a) - \ln (c)$ .

$\frac{q (\ln (a) - \ln (c))}{\ln (a) - \ln (c)} = \frac{- p \ln (c)}{\ln (a) - \ln (c)} + \frac{r \ln (a)}{\ln (a) - \ln (c)} + \frac{p \ln (b)}{\ln (a) - \ln (c)} + \frac{- r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.2

Simplify the left side.

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Step 13.2.1

Cancel the common factor of $\ln (a) - \ln (c)$ .

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Step 13.2.1.1

Cancel the common factor.

Step 13.2.1.2

Divide $q$ by $1$ .

$q = \frac{- p \ln (c)}{\ln (a) - \ln (c)} + \frac{r \ln (a)}{\ln (a) - \ln (c)} + \frac{p \ln (b)}{\ln (a) - \ln (c)} + \frac{- r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Simplify each term.

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Step 13.3.1.1

Move the negative in front of the fraction.

$q = - \frac{p \ln (c)}{\ln (a) - \ln (c)} + \frac{r \ln (a)}{\ln (a) - \ln (c)} + \frac{p \ln (b)}{\ln (a) - \ln (c)} + \frac{- r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3.1.2

Move the negative in front of the fraction.

$q = - \frac{p \ln (c)}{\ln (a) - \ln (c)} + \frac{r \ln (a)}{\ln (a) - \ln (c)} + \frac{p \ln (b)}{\ln (a) - \ln (c)} - \frac{r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3.2

Simplify terms.

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Step 13.3.2.1

Combine the numerators over the common denominator.

$q = \frac{- p \ln (c) + r \ln (a)}{\ln (a) - \ln (c)} + \frac{p \ln (b)}{\ln (a) - \ln (c)} - \frac{r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3.2.2

Combine the numerators over the common denominator.

$q = \frac{- p \ln (c) + r \ln (a) + p \ln (b)}{\ln (a) - \ln (c)} - \frac{r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3.2.3

Combine the numerators over the common denominator.

$q = \frac{- p \ln (c) + r \ln (a) + p \ln (b) - r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3.2.4

Factor $- 1$ out of $- p \ln (c)$ .

$q = \frac{- (p \ln (c)) + r \ln (a) + p \ln (b) - r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3.2.5

Factor $- 1$ out of $r \ln (a)$ .

$q = \frac{- (p \ln (c)) - (- r \ln (a)) + p \ln (b) - r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3.2.6

Factor $- 1$ out of $- (p \ln (c)) - (- r \ln (a))$ .

$q = \frac{- (p \ln (c) - r \ln (a)) + p \ln (b) - r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3.2.7

Factor $- 1$ out of $p \ln (b)$ .

$q = \frac{- (p \ln (c) - r \ln (a)) - (- p \ln (b)) - r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3.2.8

Factor $- 1$ out of $- (p \ln (c) - r \ln (a)) - (- p \ln (b))$ .

$q = \frac{- (p \ln (c) - r \ln (a) - p \ln (b)) - r \ln (b)}{\ln (a) - \ln (c)}$

Step 13.3.2.9

Factor $- 1$ out of $- r \ln (b)$ .

$q = \frac{- (p \ln (c) - r \ln (a) - p \ln (b)) - (r \ln (b))}{\ln (a) - \ln (c)}$

Step 13.3.2.10

Factor $- 1$ out of $- (p \ln (c) - r \ln (a) - p \ln (b)) - (r \ln (b))$ .

$q = \frac{- (p \ln (c) - r \ln (a) - p \ln (b) + r \ln (b))}{\ln (a) - \ln (c)}$

Step 13.3.2.11

Simplify the expression.

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Step 13.3.2.11.1

Rewrite $- (p \ln (c) - r \ln (a) - p \ln (b) + r \ln (b))$ as $- 1 (p \ln (c) - r \ln (a) - p \ln (b) + r \ln (b))$ .

$q = \frac{- 1 (p \ln (c) - r \ln (a) - p \ln (b) + r \ln (b))}{\ln (a) - \ln (c)}$

Step 13.3.2.11.2

Move the negative in front of the fraction.

$q = - \frac{p \ln (c) - r \ln (a) - p \ln (b) + r \ln (b)}{\ln (a) - \ln (c)}$