The discussion revolves around the equation log_b(a) = log_a(b) and seeks to determine the value of ab under the conditions that a and b are not equal, both are greater than zero, and neither is equal to one. The focus is on mathematical reasoning and exploration of the implications of the logarithmic relationship.
There is agreement among some participants that ab = 1 is a valid conclusion derived from the logarithmic relationship, but the discussion does not explore any alternative conclusions or unresolved points.
The discussion assumes certain conditions about the values of a and b, such as their positivity and inequality, but does not delve into potential limitations or alternative interpretations of the logarithmic properties involved.
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