Determine the probability that the element from the set {1,2,3,4,5,6,7,8} to be solution of the equation 3^(2x-6)=81 ?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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Let us find the solution of the equation 3^(2x-6) = 81

3^(2x-6) = 81

=> 3^(2x - 6) = 3^4

we can equate the exponent

2x - 6 = 4

=> x = 10/2 = 5

The number of elements in the set {1,2,3,4,5,6,7,8} are 8. The probability that 1 of them is the solution of the equation is 1/8 as 5 is one element out of 8 options.

The probability that an element from the set is a solution of the equation is 1/8

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Let's recall the formula of probability:

P = m / n, where m is the number of ways an event, that has the property "solution of the equation: 3^(2x-6)=81" can occure and n is the total number of possible outcomes.

To determine the value for m, we have to solve, at first, the equation

3^(2x-6)=81

We've noticed that 81 is a power of 3 and we'll create matching bases, writting 81 = 3^4

3^(2x-6)=3^4

Since the bases are matching, we'll apply one to one property:

2x-6=4

We'll add 6 both sides:

2x=6+4

x=10/2

x=5

Knowing that x=5 is the single root for the equation 3^(2x-6)=81, that means that m=1.

P=m/n, where m=1 and n=8 (8 countable elements in the set)

The probability of an element from the given set to be the solution of the equation 3^(2x-6)=81 is: P=1/8.

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