Precalculus Examples

$16^{n - 7} + 5 = 24$

Step 1

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

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Step 1.1

Subtract $5$ from both sides of the equation.

$16^{n - 7} = 24 - 5$

Step 1.2

Subtract $5$ from $24$ .

$16^{n - 7} = 19$

Step 2

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

$\ln (16^{n - 7}) = \ln (19)$

Step 3

Expand $\ln (16^{n - 7})$ by moving $n - 7$ outside the logarithm.

$(n - 7) \ln (16) = \ln (19)$

Step 4

Simplify the left side.

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Step 4.1

Apply the distributive property.

$n \ln (16) - 7 \ln (16) = \ln (19)$

Step 5

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

$n \ln (16) - 7 \ln (16) - \ln (19) = 0$

Step 6

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

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Step 6.1

Add $7 \ln (16)$ to both sides of the equation.

$n \ln (16) - \ln (19) = 7 \ln (16)$

Step 6.2

Add $\ln (19)$ to both sides of the equation.

$n \ln (16) = 7 \ln (16) + \ln (19)$

Step 7

Divide each term in $n \ln (16) = 7 \ln (16) + \ln (19)$ by $\ln (16)$ and simplify.

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Step 7.1

Divide each term in $n \ln (16) = 7 \ln (16) + \ln (19)$ by $\ln (16)$ .

$\frac{n \ln (16)}{\ln (16)} = \frac{7 \ln (16)}{\ln (16)} + \frac{\ln (19)}{\ln (16)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (16)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{n \ln (16)}{\ln (16)} = \frac{7 \ln (16)}{\ln (16)} + \frac{\ln (19)}{\ln (16)}$

Step 7.2.1.2

Divide $n$ by $1$ .

$n = \frac{7 \ln (16)}{\ln (16)} + \frac{\ln (19)}{\ln (16)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $\ln (16)$ .

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Step 7.3.1.1

Cancel the common factor.

$n = \frac{7 \ln (16)}{\ln (16)} + \frac{\ln (19)}{\ln (16)}$

Step 7.3.1.2

Divide $7$ by $1$ .

$n = 7 + \frac{\ln (19)}{\ln (16)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$n = 7 + \frac{\ln (19)}{\ln (16)}$

Decimal Form:

$n = 8.06198187 \dots$