SOLUTION: prove that : log 3 of base 2 multiply log 2 of base 3 = 1

Question 279629: prove that : log 3 of base 2 multiply log 2 of base 3 = 1
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
log(2,3) * log(3,2) = 1

by the conversion of logs from one base to another formula, we get:

log(2,3) = log(10,3) / log(10,2) and we get:

log(3,2) = log(10,2) / log(10,3)

substituting these equivalencies into the original equation, we get:

log(10,3) / log(10,2) * log(10,2) / log(10,3) = 1

this is equivalent to:

(log(10,3) * log(10,2)) / ((log(10,2) * log(10,3)) = 1

log(10,3) in the numerator and denominator cancel out.

log(10,2) in the numerator and denominator cancel out.

we are left with 1 = 1 which is true confirming that the identity is correct.

we can solve this another way as well.

original equation is:

log(2,3) * log(3,2) = 1

log(2,3) = y if an only if 2^y = 3

log (3,2) = z if and only if 3^z = 2

take the log of both sides of the equation of 2^y = 3 and you get:

log(2^y) = log(3)

by the rules of logarithms, this becomes:

y*log(2) = log(3)

divide both sides of this equation by log(2) to get:

y = log(3)/log(2)

use your calculator to solve for y to get y = 1.584962501 **********

now take the log of both sides of the equation of 3^z = 2 and you get:

log(3^z) = log(2)

by the rules of logarithms, this becomes:

z*log(3) = log(2)

divide both sides of this equation by log(3) to get:

z = log(2)/log(3)

use your calculator to get z = .630929754

your original equation is:

log(2,3) * log(3,2) = 1

we set log(2,3) = y and we set log(3,2) = z to get:

y * z = 1

we solved for y to get y = 1.584962501 and we solved for x to get z = .630929754.

we substitute these values for y and z in the equation to get:

1.584962501 * .630929754 = 1

we simplify to get 1 = 1 confirming that the original equation is true.

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