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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 will take the logarithm of both sides

step 2

Applying the logarithm, we get

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

step 3

Using the property of logarithms

a \log(b) = \log(b^a)

, we can rewrite the equation as

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

step 4

Now, we will solve for

x

by expanding and rearranging the terms:

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

step 5

Isolating

x

on one side gives us

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)

to solve for

x

, we get

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 use logarithms to bring the variable down to a solvable level. By applying the properties of logarithms, we can isolate the variable and solve for it.

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How to calculate the value of

x

in the equation \(2^{x-5}=3^{x+1}\

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