The one-to-one property of exponential functions states that if two exponential expressions with the same base are equal, their exponents must be equal; this allows solving exponential equations by equating exponents without using logarithms, as demonstrated by solving 5^(-n) = 125^(3n + 5) by first rewriting 125 as 5^3 to get 5^(-n) = 5^(9n + 15), then setting -n = 9n + 15 to find n = -3/2.
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One-To-One is ESSENTIAL for Solving 5^-n = 125^(3n + 5) (Solve Exponential Equation WITHOUT Logs!)Added:
Function is 5^x, it's what's called a one-to-one function; distinct inputs give distinct outputs.
Said another way: if you get the same output the inputs must have been the same. So if this equals this, then the input -n, well if you just think of it as 5 to the x, -n and 9 n + 15 must be the same: -n must equal 9n + 15. Solve for n. Add n to both sides. Subtract 15 from both sides.
-15 = 10 n. So n is -15/10, which is -3/2.
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