Solve for r 16^(r+8)-9.6=75

$16^{r + 8} - 9.6 = 75$

Step 1

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

Add $9.6$ to both sides of the equation.

$16^{r + 8} = 75 + 9.6$

Add $75$ and $9.6$ .

$16^{r + 8} = 84.6$

Step 2

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

$\ln (16^{r + 8}) = \ln (84.6)$

Step 3

Expand $\ln (16^{r + 8})$ by moving $r + 8$ outside the logarithm.

$(r + 8) \ln (16) = \ln (84.6)$

Step 4

Simplify the left side.

Apply the distributive property.

$r \ln (16) + 8 \ln (16) = \ln (84.6)$

Step 5

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

$r \ln (16) + 8 \ln (16) - \ln (84.6) = 0$

Step 6

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

Subtract $8 \ln (16)$ from both sides of the equation.

$r \ln (16) - \ln (84.6) = - 8 \ln (16)$

Add $\ln (84.6)$ to both sides of the equation.

$r \ln (16) = - 8 \ln (16) + \ln (84.6)$

Step 7

Divide each term in $r \ln (16) = - 8 \ln (16) + \ln (84.6)$ by $\ln (16)$ and simplify.

Divide each term in $r \ln (16) = - 8 \ln (16) + \ln (84.6)$ by $\ln (16)$ .

$\frac{r \ln (16)}{\ln (16)} = \frac{- 8 \ln (16)}{\ln (16)} + \frac{\ln (84.6)}{\ln (16)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{r \ln (16)}{\ln (16)} = \frac{- 8 \ln (16)}{\ln (16)} + \frac{\ln (84.6)}{\ln (16)}$

Divide $r$ by $1$ .

$r = \frac{- 8 \ln (16)}{\ln (16)} + \frac{\ln (84.6)}{\ln (16)}$

Simplify the right side.

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

Cancel the common factor.

$r = \frac{- 8 \ln (16)}{\ln (16)} + \frac{\ln (84.6)}{\ln (16)}$

Divide $- 8$ by $1$ .

$r = - 8 + \frac{\ln (84.6)}{\ln (16)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$r = - 8 + \frac{\ln (84.6)}{\ln (16)}$

Decimal Form:

$r = - 6.39935356 \dots$

Solve for r 16^(r+8)-9.6=75

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