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Solve the exponential equation for x. {:[2^(4x-5)*32^(x+1)=2^(6x-7)],[x=◻]:}

Solve the exponential equation for x x .\newline24x−5⋅32x+1=26x−7x=□ \begin{array}{l} 2^{4 x-5} \cdot 32^{x+1}=2^{6 x-7} \\ x=\square \end{array}

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Q. Solve the exponential equation for x x .\newline24x−5⋅32x+1=26x−7x=□ \begin{array}{l} 2^{4 x-5} \cdot 32^{x+1}=2^{6 x-7} \\ x=\square \end{array}
  1. Write Equation: Write down the given exponential equation and simplify the bases to have the same base.\newlineThe given equation is:\newline24x−5×32x+1=26x−72^{4x-5} \times 32^{x+1} = 2^{6x-7}\newlineWe know that 3232 is 252^5, so we can rewrite 32x+132^{x+1} as (25)x+1(2^5)^{x+1}.
  2. Apply Power Rule: Apply the power of a power rule to simplify the equation.\newlineUsing the power of a power rule (ab)c=ab∗c(a^b)^c = a^{b*c}, we get:\newline24x−5×(25)x+1=26x−72^{4x-5} \times (2^5)^{x+1} = 2^{6x-7}\newline24x−5×25x+5=26x−72^{4x-5} \times 2^{5x+5} = 2^{6x-7}
  3. Combine Terms: Combine the terms with the same base on the left side of the equation using the property of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}. \newline24x−5+5x+5=26x−72^{4x-5 + 5x+5} = 2^{6x-7}\newline29x=26x−72^{9x} = 2^{6x-7}
  4. Set Exponents Equal: Since the bases are the same, we can set the exponents equal to each other and solve for xx.9x=6x−79x = 6x - 7
  5. Subtract and Isolate: Subtract 6x6x from both sides to isolate the variable on one side.\newline9x−6x=6x−6x−79x - 6x = 6x - 6x - 7\newline3x=−73x = -7
  6. Divide and Solve: Divide both sides by 33 to solve for xx.3x3=−73\frac{3x}{3} = \frac{-7}{3}x=−73x = \frac{-7}{3}

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