Logarithms help???

Question

find the value of,
3^{ SQR( log 7 / log 3) } - 7^{SQR(log 3 / log 7)}
Thank you in advance

How to write the symbol SQR?

Answer 1

Although the first person to answer has come to the right conclusion, they have made two errors in their working.

1. You must not start by equating things until you know that they are equal. It is well known that you can start with a false equality and end up with a true one by correct mathematical operation. The most obvious example is 2 = -2 square both sides 4 = 4.

2. In the middle, when they squared both sides, the left hand side should have been (log7/log3)*(log3)^2 with something similar on the other side. Now it is true that log(3^2) = 2*log3 but (log3)^2 is NOT the same as 2*log3. (Check it on a calculator!)

It is just good luck that neither of these errors affected the answer. The correct way to start, if you suspect that the two terms are equal is to say let 3^{SQR(log7/log3)} = (7^x) and then prove that x = SQR(log3/log7).

Answer 2

3^{ SQR( log 7 / log 3) }=7^{SQR(log 3 / log 7)}
Take the log of both sides and using a property of logs
{ SQR( log 7 / log 3) } * log 3 ={SQR(log 3 / log 7)} * log 7
Then square both sides
(log 7 / log 3) * 2 log 3 = (log 3 / log 7) * 2 log 7
It's 2 log 3 above because of the property of logs
Then start cancelling
log 7 log 3 = log 3 log 7

Basically this tells you that they are equal to one another so the value of the expression you asked is 0.