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Find the expression for f(x) that makes the following equation true for all values of x. {:[(81^(x))/(9^(5x-8))=9^(f(x))],[f(x)=◻]:}

Find the expression for f(x) f(x) that makes the following equation true for all values of x x .\newline81x95x−8=9f(x)f(x)=□ \begin{array}{l} \frac{81^{x}}{9^{5 x-8}}=9^{f(x)} \\ f(x)=\square \end{array}

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Q. Find the expression for f(x) f(x) that makes the following equation true for all values of x x .\newline81x95x−8=9f(x)f(x)=□ \begin{array}{l} \frac{81^{x}}{9^{5 x-8}}=9^{f(x)} \\ f(x)=\square \end{array}
  1. Simplify left side using exponents: First, let's simplify the left side of the equation using properties of exponents. We know that 8181 is 99 squared (929^2), so we can rewrite 81x81^{x} as (92)x(9^2)^{x}.
  2. Apply power of a power rule: Now, apply the power of a power rule, which states that (ab)c=a(b∗c)(a^b)^c = a^{(b*c)}. So, (92)x(9^2)^x becomes 92x9^{2x}.
  3. Divide and simplify using quotient rule: Next, we divide 92x9^{2x} by 95x−89^{5x-8}. According to the quotient rule for exponents, am/an=am−na^{m}/a^{n} = a^{m-n}. Therefore, 92x/95x−89^{2x} / 9^{5x-8} simplifies to 92x−(5x−8)9^{2x - (5x-8)}.
  4. Distribute and simplify exponent: Now, let's simplify the exponent by distributing the negative sign inside the parentheses: 2x−(5x−8)2x - (5x-8) becomes 2x−5x+82x - 5x + 8.
  5. Combine like terms: Simplify the expression further by combining like terms: 2x−5x+82x - 5x + 8 simplifies to −3x+8-3x + 8.
  6. Determine f(x)f(x): Now we have the left side of the equation simplified to 9(−3x+8)9^{(-3x + 8)}. According to the original equation, this must be equal to 9(f(x))9^{(f(x))}. Therefore, f(x)f(x) must be equal to the exponent we found, which is −3x+8-3x + 8.

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