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

Jan 30, 2024

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

Generated Graph

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 simplify to

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

step 4

Rearrange the equation to isolate terms with

x

on one side:

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

step 5

Group the

x

terms together and the constant terms together:

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

step 6

Solve for

x

by dividing both sides by

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

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

Solving Exponential Equations Using Logarithms

Explanation

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