maths...

Went to play pool with Darren and YH yesterday. After that went Karaoke with Darren alone. Darren has a better voice than me. Damn.

There was a Math problem last post.

a^x = -1
a^2x = 1
a^2x = a^0
2x = 0
x = 0

sub into first equation

a^0 = -1
1 = -1

Okay the error is really simple.

In secondary school you probably learnt that
if a^x = a^y, then x=y
this is, in general, not true.

For eq,
(-1)^6=1
(-1)^8=1

Thus (-1)^6 = (-1)^8
But that certainly does not mean that 6=8.

In general, if a^x=a^y
then y=x if you can take the logarithm of both sides and reduce the expression to x=y.

For example in the first example of this post,

a^x = -1

From this, it follows that a is negative, since a positive number to any real exponent would be positive.

then a^2x = 1
a^2x = a^0

But since a is negative, we certainly can't take the logarithm of it and thus the equation cannot reduce to 2x=0.

You might be wondering, how about the equation
1^6=1^8

In this case we can take the logarithm of both sides, since ln1 is defined, but doesn't that mean that 6=8?

Well no, lets do out the formal maths

1^6 = 1^8
ln(1^6) = ln(1^8)

bringing down the powers,

6*ln1 = 8*ln1

Realize that at this point, we cannot simply cancel out the ln1's because ln1 is zero. Doing so will mean doing a division by zero, which is simply undefined in mathematics.

In other words,

6*0=8*0

doesn't mean that 6=8.