Precalculus Examples

$6 \cdot 11^{3 n - 9} + 10 = 47$

Step 1

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

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Step 1.1

Subtract $10$ from both sides of the equation.

$6 \cdot 11^{3 n - 9} = 47 - 10$

Step 1.2

Subtract $10$ from $47$ .

$6 \cdot 11^{3 n - 9} = 37$

Step 2

Divide each term in $6 \cdot 11^{3 n - 9} = 37$ by $6$ and simplify.

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Step 2.1

Divide each term in $6 \cdot 11^{3 n - 9} = 37$ by $6$ .

$\frac{6 \cdot 11^{3 n - 9}}{6} = \frac{37}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{6 \cdot 11^{3 n - 9}}{6} = \frac{37}{6}$

Step 2.2.1.2

Divide $11^{3 n - 9}$ by $1$ .

$11^{3 n - 9} = \frac{37}{6}$

Step 3

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

$\ln (11^{3 n - 9}) = \ln (\frac{37}{6})$

Step 4

Expand $\ln (11^{3 n - 9})$ by moving $3 n - 9$ outside the logarithm.

$(3 n - 9) \ln (11) = \ln (\frac{37}{6})$

Step 5

Simplify the left side.

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Step 5.1

Apply the distributive property.

$3 n \ln (11) - 9 \ln (11) = \ln (\frac{37}{6})$

Step 6

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

$3 n \ln (11) - 9 \ln (11) - \ln (\frac{37}{6}) = 0$

Step 7

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

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Step 7.1

Add $9 \ln (11)$ to both sides of the equation.

$3 n \ln (11) - \ln (\frac{37}{6}) = 9 \ln (11)$

Step 7.2

Add $\ln (\frac{37}{6})$ to both sides of the equation.

$3 n \ln (11) = 9 \ln (11) + \ln (\frac{37}{6})$

Step 8

Divide each term in $3 n \ln (11) = 9 \ln (11) + \ln (\frac{37}{6})$ by $3 \ln (11)$ and simplify.

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Step 8.1

Divide each term in $3 n \ln (11) = 9 \ln (11) + \ln (\frac{37}{6})$ by $3 \ln (11)$ .

$\frac{3 n \ln (11)}{3 \ln (11)} = \frac{9 \ln (11)}{3 \ln (11)} + \frac{\ln (\frac{37}{6})}{3 \ln (11)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $3$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{3 n \ln (11)}{3 \ln (11)} = \frac{9 \ln (11)}{3 \ln (11)} + \frac{\ln (\frac{37}{6})}{3 \ln (11)}$

Step 8.2.1.2

Rewrite the expression.

$\frac{n \ln (11)}{\ln (11)} = \frac{9 \ln (11)}{3 \ln (11)} + \frac{\ln (\frac{37}{6})}{3 \ln (11)}$

Step 8.2.2

Cancel the common factor of $\ln (11)$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{n \ln (11)}{\ln (11)} = \frac{9 \ln (11)}{3 \ln (11)} + \frac{\ln (\frac{37}{6})}{3 \ln (11)}$

Step 8.2.2.2

Divide $n$ by $1$ .

$n = \frac{9 \ln (11)}{3 \ln (11)} + \frac{\ln (\frac{37}{6})}{3 \ln (11)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $9$ and $3$ .

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Step 8.3.1.1

Factor $3$ out of $9 \ln (11)$ .

$n = \frac{3 (3 \ln (11))}{3 \ln (11)} + \frac{\ln (\frac{37}{6})}{3 \ln (11)}$

Step 8.3.1.2

Cancel the common factor.

$n = \frac{3 (3 \ln (11))}{3 \ln (11)} + \frac{\ln (\frac{37}{6})}{3 \ln (11)}$

Step 8.3.1.3

Rewrite the expression.

$n = 3 + \frac{\ln (\frac{37}{6})}{3 \ln (11)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$n = 3 + \frac{\ln (\frac{37}{6})}{3 \ln (11)}$

Decimal Form:

$n = 3.25288266 \dots$