Solve for r 9*12^(r+4)-8=127

$9 \cdot 12^{r + 4} - 8 = 127$

Step 1

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

Add $8$ to both sides of the equation.

$9 \cdot 12^{r + 4} = 127 + 8$

Add $127$ and $8$ .

$9 \cdot 12^{r + 4} = 135$

Step 2

Divide each term in $(9 \cdot 12^{r + 4}) = 135$ by $9$ and simplify.

Divide each term in $(9 \cdot 12^{r + 4}) = 135$ by $9$ .

$\frac{(9 \cdot 12^{r + 4})}{9} = \frac{135}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 \cdot 12^{r + 4}}{9} = \frac{135}{9}$

Divide $12^{r + 4}$ by $1$ .

$12^{r + 4} = \frac{135}{9}$

Simplify the right side.

Divide $135$ by $9$ .

$12^{r + 4} = 15$

Step 3

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

$\ln (12^{r + 4}) = \ln (15)$

Step 4

Expand $\ln (12^{r + 4})$ by moving $r + 4$ outside the logarithm.

$(r + 4) \ln (12) = \ln (15)$

Step 5

Simplify the left side.

Apply the distributive property.

$r \ln (12) + 4 \ln (12) = \ln (15)$

Step 6

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

$r \ln (12) + 4 \ln (12) - \ln (15) = 0$

Step 7

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

Subtract $4 \ln (12)$ from both sides of the equation.

$r \ln (12) - \ln (15) = - 4 \ln (12)$

Add $\ln (15)$ to both sides of the equation.

$r \ln (12) = - 4 \ln (12) + \ln (15)$

Step 8

Divide each term in $r \ln (12) = - 4 \ln (12) + \ln (15)$ by $\ln (12)$ and simplify.

Divide each term in $r \ln (12) = - 4 \ln (12) + \ln (15)$ by $\ln (12)$ .

$\frac{r \ln (12)}{\ln (12)} = \frac{- 4 \ln (12)}{\ln (12)} + \frac{\ln (15)}{\ln (12)}$

Simplify the left side.

Cancel the common factor of $\ln (12)$ .

Cancel the common factor.

$\frac{r \ln (12)}{\ln (12)} = \frac{- 4 \ln (12)}{\ln (12)} + \frac{\ln (15)}{\ln (12)}$

Divide $r$ by $1$ .

$r = \frac{- 4 \ln (12)}{\ln (12)} + \frac{\ln (15)}{\ln (12)}$

Simplify the right side.

Cancel the common factor of $\ln (12)$ .

Cancel the common factor.

$r = \frac{- 4 \ln (12)}{\ln (12)} + \frac{\ln (15)}{\ln (12)}$

Divide $- 4$ by $1$ .

$r = - 4 + \frac{\ln (15)}{\ln (12)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$r = - 4 + \frac{\ln (15)}{\ln (12)}$

Decimal Form:

$r = - 2.91020042 \dots$

Solve for r 9*12^(r+4)-8=127

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