Solve for x (81^x)/(9^(5x-8))=9^(f(x))

$\frac{81^{x}}{9^{5 x - 8}} = 9^{f (x)}$

Move $9^{5 x - 8}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$81^{x} \cdot 9^{- (5 x - 8)} = 9^{f (x)}$

Rewrite $81$ as $3^{4}$ .

${(3^{4})}^{x} \cdot 9^{- (5 x - 8)} = 9^{f (x)}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3^{4 x} \cdot 9^{- (5 x - 8)} = 9^{f (x)}$

Rewrite $9$ as $3^{2}$ .

$3^{4 x} \cdot {(3^{2})}^{- (5 x - 8)} = 9^{f (x)}$

Multiply the exponents in ${(3^{2})}^{- (5 x - 8)}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3^{4 x} \cdot 3^{2 (- (5 x - 8))} = 9^{f (x)}$

Apply the distributive property.

$3^{4 x} \cdot 3^{2 (- (5 x) - - 8)} = 9^{f (x)}$

Multiply $5$ by $- 1$ .

$3^{4 x} \cdot 3^{2 (- 5 x - - 8)} = 9^{f (x)}$

Multiply $- 1$ by $- 8$ .

$3^{4 x} \cdot 3^{2 (- 5 x + 8)} = 9^{f (x)}$

Apply the distributive property.

$3^{4 x} \cdot 3^{2 (- 5 x) + 2 \cdot 8} = 9^{f (x)}$

Multiply $- 5$ by $2$ .

$3^{4 x} \cdot 3^{- 10 x + 2 \cdot 8} = 9^{f (x)}$

Multiply $2$ by $8$ .

$3^{4 x} \cdot 3^{- 10 x + 16} = 9^{f (x)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3^{4 x - 10 x + 16} = 9^{f (x)}$

Subtract $10 x$ from $4 x$ .

$3^{- 6 x + 16} = 9^{f (x)}$

Create equivalent expressions in the equation that all have equal bases.

$3^{- 6 x + 16} = 3^{2 f (x)}$

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$- 6 x + 16 = 2 f (x)$

Solve for $x$ .

Subtract $2 f x$ from both sides of the equation.

$- 6 x + 16 - 2 f x = 0$

Subtract $16$ from both sides of the equation.

$- 6 x - 2 f x = - 16$

Factor $2 x$ out of $- 6 x - 2 f x$ .

Factor $2 x$ out of $- 6 x$ .

$2 x (- 3) - 2 f x = - 16$

Factor $2 x$ out of $- 2 f x$ .

$2 x (- 3) + 2 x (- f) = - 16$

Factor $2 x$ out of $2 x (- 3) + 2 x (- f)$ .

$2 x (- 3 - f) = - 16$

Divide each term in $2 x (- 3 - f) = - 16$ by $2 (- 3 - f)$ and simplify.

Divide each term in $2 x (- 3 - f) = - 16$ by $2 (- 3 - f)$ .

$\frac{2 x (- 3 - f)}{2 (- 3 - f)} = \frac{- 16}{2 (- 3 - f)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x (- 3 - f)}{2 (- 3 - f)} = \frac{- 16}{2 (- 3 - f)}$

Rewrite the expression.

$\frac{x (- 3 - f)}{- 3 - f} = \frac{- 16}{2 (- 3 - f)}$

Cancel the common factor of $- 3 - f$ .

Cancel the common factor.

$\frac{x (- 3 - f)}{- 3 - f} = \frac{- 16}{2 (- 3 - f)}$

Divide $x$ by $1$ .

$x = \frac{- 16}{2 (- 3 - f)}$

Simplify the right side.

Cancel the common factor of $- 16$ and $2$ .

Factor $2$ out of $- 16$ .

$x = \frac{2 \cdot - 8}{2 (- 3 - f)}$

Cancel the common factors.

Cancel the common factor.

$x = \frac{2 \cdot - 8}{2 (- 3 - f)}$

Rewrite the expression.

$x = \frac{- 8}{- 3 - f}$

Move the negative in front of the fraction.

$x = - \frac{8}{- 3 - f}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = - \frac{8}{- 1 (3) - f}$

Factor $- 1$ out of $- f$ .

$x = - \frac{8}{- 1 (3) - (f)}$

Factor $- 1$ out of $- 1 (3) - (f)$ .

$x = - \frac{8}{- 1 (3 + f)}$

Simplify the expression.

Move the negative in front of the fraction.

$x = - - \frac{8}{3 + f}$

Multiply $- 1$ by $- 1$ .

$x = 1 \frac{8}{3 + f}$

Multiply $\frac{8}{3 + f}$ by $1$ .

$x = \frac{8}{3 + f}$

Solve for x (81^x)/(9^(5x-8))=9^(f(x))

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