MHB Find ab if log_b(a) = log_a(b)

[My Name is Earl]

I have tried various methods to solve this...

If log_b(a) = log_a(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]

I have tried various methods to solve this...

If log_b(a) = log_a(b) where a != b (!= means does not equal), ab > 0 and neither a nor b are 1, then what is the value of ab?

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]

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?

This implies that ab = 1

 

[MarkFL]

This implies that ab = 1

Yes, that's what I found as well. (Yes)