Solve for x 14^(3-8x)+9=77

$14^{3 - 8 x} + 9 = 77$

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

Subtract $9$ from both sides of the equation.

$14^{3 - 8 x} = 77 - 9$

Subtract $9$ from $77$ .

$14^{3 - 8 x} = 68$

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

$\ln (14^{3 - 8 x}) = \ln (68)$

Expand $\ln (14^{3 - 8 x})$ by moving $3 - 8 x$ outside the logarithm.

$(3 - 8 x) \ln (14) = \ln (68)$

Simplify the left side.

Apply the distributive property.

$3 \ln (14) - 8 x \ln (14) = \ln (68)$

Reorder $3 \ln (14)$ and $- 8 x \ln (14)$ .

$- 8 x \ln (14) + 3 \ln (14) = \ln (68)$

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

$- 8 x \ln (14) + 3 \ln (14) - \ln (68) = 0$

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

Subtract $3 \ln (14)$ from both sides of the equation.

$- 8 x \ln (14) - \ln (68) = - 3 \ln (14)$

Add $\ln (68)$ to both sides of the equation.

$- 8 x \ln (14) = - 3 \ln (14) + \ln (68)$

Divide each term in $- 8 x \ln (14) = - 3 \ln (14) + \ln (68)$ by $- 8 \ln (14)$ and simplify.

Divide each term in $- 8 x \ln (14) = - 3 \ln (14) + \ln (68)$ by $- 8 \ln (14)$ .

$\frac{- 8 x \ln (14)}{- 8 \ln (14)} = \frac{- 3 \ln (14)}{- 8 \ln (14)} + \frac{\ln (68)}{- 8 \ln (14)}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 x \ln (14)}{- 8 \ln (14)} = \frac{- 3 \ln (14)}{- 8 \ln (14)} + \frac{\ln (68)}{- 8 \ln (14)}$

Rewrite the expression.

$\frac{x \ln (14)}{\ln (14)} = \frac{- 3 \ln (14)}{- 8 \ln (14)} + \frac{\ln (68)}{- 8 \ln (14)}$

Cancel the common factor of $\ln (14)$ .

Cancel the common factor.

$\frac{x \ln (14)}{\ln (14)} = \frac{- 3 \ln (14)}{- 8 \ln (14)} + \frac{\ln (68)}{- 8 \ln (14)}$

Divide $x$ by $1$ .

$x = \frac{- 3 \ln (14)}{- 8 \ln (14)} + \frac{\ln (68)}{- 8 \ln (14)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $\ln (14)$ .

Cancel the common factor.

$x = \frac{- 3 \ln (14)}{- 8 \ln (14)} + \frac{\ln (68)}{- 8 \ln (14)}$

Rewrite the expression.

$x = \frac{- 3}{- 8} + \frac{\ln (68)}{- 8 \ln (14)}$

Dividing two negative values results in a positive value.

$x = \frac{3}{8} + \frac{\ln (68)}{- 8 \ln (14)}$

Move the negative in front of the fraction.

$x = \frac{3}{8} - \frac{\ln (68)}{8 \ln (14)}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{8} - \frac{\ln (68)}{8 \ln (14)}$

Decimal Form:

$x = 0.17514133 \dots$

Solve for x 14^(3-8x)+9=77

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