How to use the property of exponentials of being injective, in an equation?
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Here's an example:
4^(x-1) = 8*2^3x , x = ?
Since 8 is a power of 2, we'll write it as:
Now, we'll apply the multiplication rule of 2 exponentials that have matching bases:
2^3*2^3x = 2^(3 + 3x)
We'll re-write the equation, putting 4 = 2^2
2^2(x-1) = 2^(3 + 3x)
Since the bases are matching, we'll apply one to one rule:
2(x-1) = (3 + 3x)
We'll open the brackets:
2x - 2 = 3x + 3
We'll subtract 2x - 2 and we'll apply symmetrical property:
3x + 3 - 2x + 2 = 0
x + 5 = 0
We'll subtract 5:
x = -5
The solution of the equation is x = -5.
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