The equation log_b(a) = log_a(b) leads to the conclusion that the product ab equals 1, given the conditions that a and b are not equal, both are greater than 0, and neither is equal to 1. By letting x = log_a(b) = log_b(a), it follows that a^x = b and b^x = a. Dividing these equations results in the expression (a/b)^x = (a/b)^{-1}, confirming that ab = 1.
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My Name is Earl said:I have tried various methods to solve this...
If logb(a) = loga(b) where a != b (!= means does not equal), ab > 0 and neither a nor b are 1, then what is the value of ab?
MarkFL said:Let's let:
$$x=\log_a(b)=\log_b(a)$$
Now this implies:
$$a^x=b$$
$$b^x=a$$
Dividing the former by the latter, we obtain:
$$\left(\frac{a}{b}\right)^x=\left(\frac{a}{b}\right)^{-1}$$
What does this imply?
My Name is Earl said:This implies that ab = 1