Questions: Solve the given equation for x. 2^(4-5x) = 2^(2x-6)

Transcript text: Solve the given equation for x . \[ 2^{4-5 x}=2^{2 x-6} \]

Solution

Solution Steps

To solve the equation \(2^{4-5x} = 2^{2x-6}\), we can use the property of exponents that states if the bases are the same, then the exponents must be equal. Therefore, we set the exponents equal to each other and solve for \(x\).

Step 1: Equate the Exponents

Given the equation \(2^{4-5x} = 2^{2x-6}\), we use the property of exponents that states if the bases are the same, the exponents must be equal. Therefore, we equate the exponents: \[ 4 - 5x = 2x - 6 \]

Step 2: Solve for \(x\)

To solve for \(x\), we rearrange the equation: \[ 4 + 6 = 2x + 5x \] \[ 10 = 7x \]

Step 3: Isolate \(x\)

Divide both sides by 7 to isolate \(x\): \[ x = \frac{10}{7} \]