Precalculus Examples

$9^{n + 10} + 3 = 81$

Step 1

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

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Step 1.1

Subtract $3$ from both sides of the equation.

$9^{n + 10} = 81 - 3$

Step 1.2

Subtract $3$ from $81$ .

$9^{n + 10} = 78$

Step 2

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

$\ln (9^{n + 10}) = \ln (78)$

Step 3

Expand $\ln (9^{n + 10})$ by moving $n + 10$ outside the logarithm.

$(n + 10) \ln (9) = \ln (78)$

Step 4

Simplify the left side.

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Step 4.1

Apply the distributive property.

$n \ln (9) + 10 \ln (9) = \ln (78)$

Step 5

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

$n \ln (9) + 10 \ln (9) - \ln (78) = 0$

Step 6

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

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Step 6.1

Subtract $10 \ln (9)$ from both sides of the equation.

$n \ln (9) - \ln (78) = - 10 \ln (9)$

Step 6.2

Add $\ln (78)$ to both sides of the equation.

$n \ln (9) = - 10 \ln (9) + \ln (78)$

Step 7

Divide each term in $n \ln (9) = - 10 \ln (9) + \ln (78)$ by $\ln (9)$ and simplify.

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Step 7.1

Divide each term in $n \ln (9) = - 10 \ln (9) + \ln (78)$ by $\ln (9)$ .

$\frac{n \ln (9)}{\ln (9)} = \frac{- 10 \ln (9)}{\ln (9)} + \frac{\ln (78)}{\ln (9)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (9)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{n \ln (9)}{\ln (9)} = \frac{- 10 \ln (9)}{\ln (9)} + \frac{\ln (78)}{\ln (9)}$

Step 7.2.1.2

Divide $n$ by $1$ .

$n = \frac{- 10 \ln (9)}{\ln (9)} + \frac{\ln (78)}{\ln (9)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $\ln (9)$ .

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Step 7.3.1.1

Cancel the common factor.

$n = \frac{- 10 \ln (9)}{\ln (9)} + \frac{\ln (78)}{\ln (9)}$

Step 7.3.1.2

Divide $- 10$ by $1$ .

$n = - 10 + \frac{\ln (78)}{\ln (9)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$n = - 10 + \frac{\ln (78)}{\ln (9)}$

Decimal Form:

$n = - 8.01717636 \dots$