Precalculus Examples

$- 10 \cdot 20^{5 - 8 n} + 7 = - 75$

Step 1

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

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Step 1.1

Subtract $7$ from both sides of the equation.

$- 10 \cdot 20^{5 - 8 n} = - 75 - 7$

Step 1.2

Subtract $7$ from $- 75$ .

$- 10 \cdot 20^{5 - 8 n} = - 82$

Step 2

Divide each term in $- 10 \cdot 20^{5 - 8 n} = - 82$ by $- 10$ and simplify.

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Step 2.1

Divide each term in $- 10 \cdot 20^{5 - 8 n} = - 82$ by $- 10$ .

$\frac{- 10 \cdot 20^{5 - 8 n}}{- 10} = \frac{- 82}{- 10}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 10$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 10 \cdot 20^{5 - 8 n}}{- 10} = \frac{- 82}{- 10}$

Step 2.2.1.2

Divide $20^{5 - 8 n}$ by $1$ .

$20^{5 - 8 n} = \frac{- 82}{- 10}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $- 82$ and $- 10$ .

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Step 2.3.1.1

Factor $- 2$ out of $- 82$ .

$20^{5 - 8 n} = \frac{- 2 (41)}{- 10}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $- 2$ out of $- 10$ .

$20^{5 - 8 n} = \frac{- 2 \cdot 41}{- 2 \cdot 5}$

Step 2.3.1.2.2

Cancel the common factor.

$20^{5 - 8 n} = \frac{- 2 \cdot 41}{- 2 \cdot 5}$

Step 2.3.1.2.3

Rewrite the expression.

$20^{5 - 8 n} = \frac{41}{5}$

Step 3

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

$\ln (20^{5 - 8 n}) = \ln (\frac{41}{5})$

Step 4

Expand $\ln (20^{5 - 8 n})$ by moving $5 - 8 n$ outside the logarithm.

$(5 - 8 n) \ln (20) = \ln (\frac{41}{5})$

Step 5

Simplify the left side.

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Step 5.1

Apply the distributive property.

$5 \ln (20) - 8 n \ln (20) = \ln (\frac{41}{5})$

Step 6

Reorder $5 \ln (20)$ and $- 8 n \ln (20)$ .

$- 8 n \ln (20) + 5 \ln (20) = \ln (\frac{41}{5})$

Step 7

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

$- 8 n \ln (20) + 5 \ln (20) - \ln (\frac{41}{5}) = 0$

Step 8

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

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Step 8.1

Subtract $5 \ln (20)$ from both sides of the equation.

$- 8 n \ln (20) - \ln (\frac{41}{5}) = - 5 \ln (20)$

Step 8.2

Add $\ln (\frac{41}{5})$ to both sides of the equation.

$- 8 n \ln (20) = - 5 \ln (20) + \ln (\frac{41}{5})$

Step 9

Divide each term in $- 8 n \ln (20) = - 5 \ln (20) + \ln (\frac{41}{5})$ by $- 8 \ln (20)$ and simplify.

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Step 9.1

Divide each term in $- 8 n \ln (20) = - 5 \ln (20) + \ln (\frac{41}{5})$ by $- 8 \ln (20)$ .

$\frac{- 8 n \ln (20)}{- 8 \ln (20)} = \frac{- 5 \ln (20)}{- 8 \ln (20)} + \frac{\ln (\frac{41}{5})}{- 8 \ln (20)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 8$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 8 n \ln (20)}{- 8 \ln (20)} = \frac{- 5 \ln (20)}{- 8 \ln (20)} + \frac{\ln (\frac{41}{5})}{- 8 \ln (20)}$

Step 9.2.1.2

Rewrite the expression.

$\frac{n \ln (20)}{\ln (20)} = \frac{- 5 \ln (20)}{- 8 \ln (20)} + \frac{\ln (\frac{41}{5})}{- 8 \ln (20)}$

Step 9.2.2

Cancel the common factor of $\ln (20)$ .

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Step 9.2.2.1

Cancel the common factor.

$\frac{n \ln (20)}{\ln (20)} = \frac{- 5 \ln (20)}{- 8 \ln (20)} + \frac{\ln (\frac{41}{5})}{- 8 \ln (20)}$

Step 9.2.2.2

Divide $n$ by $1$ .

$n = \frac{- 5 \ln (20)}{- 8 \ln (20)} + \frac{\ln (\frac{41}{5})}{- 8 \ln (20)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

Cancel the common factor of $\ln (20)$ .

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Step 9.3.1.1.1

Cancel the common factor.

$n = \frac{- 5 \ln (20)}{- 8 \ln (20)} + \frac{\ln (\frac{41}{5})}{- 8 \ln (20)}$

Step 9.3.1.1.2

Rewrite the expression.

$n = \frac{- 5}{- 8} + \frac{\ln (\frac{41}{5})}{- 8 \ln (20)}$

Step 9.3.1.2

Dividing two negative values results in a positive value.

$n = \frac{5}{8} + \frac{\ln (\frac{41}{5})}{- 8 \ln (20)}$

Step 9.3.1.3

Move the negative in front of the fraction.

$n = \frac{5}{8} - \frac{\ln (\frac{41}{5})}{8 \ln (20)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$n = \frac{5}{8} - \frac{\ln (\frac{41}{5})}{8 \ln (20)}$

Decimal Form:

$n = 0.53720284 \dots$