Solve for c 13^(c-8)-9=17

$13^{c - 8} - 9 = 17$

Step 1

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

Add $9$ to both sides of the equation.

$13^{c - 8} = 17 + 9$

Add $17$ and $9$ .

$13^{c - 8} = 26$

Step 2

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

$\ln (13^{c - 8}) = \ln (26)$

Step 3

Expand $\ln (13^{c - 8})$ by moving $c - 8$ outside the logarithm.

$(c - 8) \ln (13) = \ln (26)$

Step 4

Simplify the left side.

Apply the distributive property.

$c \ln (13) - 8 \ln (13) = \ln (26)$

Step 5

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

$c \ln (13) - 8 \ln (13) - \ln (26) = 0$

Step 6

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

Add $8 \ln (13)$ to both sides of the equation.

$c \ln (13) - \ln (26) = 8 \ln (13)$

Add $\ln (26)$ to both sides of the equation.

$c \ln (13) = 8 \ln (13) + \ln (26)$

Step 7

Divide each term in $c \ln (13) = 8 \ln (13) + \ln (26)$ by $\ln (13)$ and simplify.

Divide each term in $c \ln (13) = 8 \ln (13) + \ln (26)$ by $\ln (13)$ .

$\frac{c \ln (13)}{\ln (13)} = \frac{8 \ln (13)}{\ln (13)} + \frac{\ln (26)}{\ln (13)}$

Simplify the left side.

Cancel the common factor of $\ln (13)$ .

Cancel the common factor.

$\frac{c \ln (13)}{\ln (13)} = \frac{8 \ln (13)}{\ln (13)} + \frac{\ln (26)}{\ln (13)}$

Divide $c$ by $1$ .

$c = \frac{8 \ln (13)}{\ln (13)} + \frac{\ln (26)}{\ln (13)}$

Simplify the right side.

Cancel the common factor of $\ln (13)$ .

Cancel the common factor.

$c = \frac{8 \ln (13)}{\ln (13)} + \frac{\ln (26)}{\ln (13)}$

Divide $8$ by $1$ .

$c = 8 + \frac{\ln (26)}{\ln (13)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$c = 8 + \frac{\ln (26)}{\ln (13)}$

Decimal Form:

$c = 9.27023815 \dots$

Solve for c 13^(c-8)-9=17

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