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Suppose \(5^{3x+2} = 675\). Find \(5^{x}\).

 May 12, 2024
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First, we try to rewrite 675 as a power of 5. We can see that 675=3⋅3⋅3⋅3⋅5=54⋅33. Since we don't care about the factor of 3, this shows that 675 can also be written as 54.

 

We now have the equation 53x+2=54. Since for any real numbers a and b with a=0, am=an implies m=n, we must have 3x+2=4. Solving for x, we find x = 2/3.

 

Alternatively, since we want to isolate 5x, taking the logarithm of both sides (with an appropriate base) would be another approach. Note that this would ultimately lead to the same solution.

 May 12, 2024

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