# Prove? #log_x(y)xxlog_y(x)=1#

##### 3 Answers

See explanation.

#### Explanation:

The change of base formula says

Applying that here we have:

By definition of the logarithm we have:

# log_a b = c iff a^c=b #

If we apply the definition to

# log_xy = Aiff x^A= y#

And if we apply the definition to

# log_yx =B iff y^B = x #

And if we replace

# (y^B)^A= y iff y^(AB)= y iff AB =1 #

And the result follows, as:

# AB =1 iff log_xy \ log_yx =B = 1# QED

See below

#### Explanation:

There are a couple of ways to do this:

- Use the log base switch rule:
#log_a(b)=1/log_b(a)#

- Use the log base change rule:
#log_a(b)=log_x(b)/log_x(a)#