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2^{x-5}=3^{x+1}

Jan 29, 2024

2^{x-5}=3^{x+1}

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Solution by Steps

step 1

To solve the equation

2^{x-5}=3^{x+1}

, we need to find a common base or use logarithms

step 2

Taking the logarithm of both sides of the equation, we get

\log(2^{x-5})=\log(3^{x+1})

step 3

Applying the power rule of logarithms, we can rewrite the equation as

(x-5)\log(2)=(x+1)\log(3)

step 4

Expanding both sides gives

x\log(2) - 5\log(2) = x\log(3) + \log(3)

step 5

Rearranging the terms to isolate x, we get

x\log(2) - x\log(3) = 5\log(2) + \log(3)

step 6

Factoring out x from the left side, we have

x(\log(2) - \log(3)) = 5\log(2) + \log(3)

step 7

Dividing both sides by

(\log(2) - \log(3))

, we find

x = \frac{5\log(2) + \log(3)}{\log(2) - \log(3)}

Answer

x = \frac{5\log(2) + \log(3)}{\log(2) - \log(3)}

Key Concept

Logarithmic Equations

Explanation

To solve an equation where the variable is in the exponent, we use logarithms to bring the variable down to a solvable form. By applying the properties of logarithms, we can isolate the variable and solve for it.