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Hi everyone. Here's a simple problem I need help with:

Find x such that [itex]12^x = 18[/itex]

From one point of view, [itex]x = log(18)/log(12)[/itex] and the problem is solved.

However, if we write 12 as [itex]3*2^2[/itex] and 18 as [itex]3^2*2[/itex] then,

[itex](2 * 3^2) = 3^x * 2^{2x}[/itex]

and hence by the uniqueness of prime factorization (in particular that of the exponents of the prime factors),

x = 2

and 2x = 1

but these equations do not have a consistent solution. I think the error is in the second reasoning.

Can someone help please?

Cheers

Vivek

Find x such that [itex]12^x = 18[/itex]

From one point of view, [itex]x = log(18)/log(12)[/itex] and the problem is solved.

However, if we write 12 as [itex]3*2^2[/itex] and 18 as [itex]3^2*2[/itex] then,

[itex](2 * 3^2) = 3^x * 2^{2x}[/itex]

and hence by the uniqueness of prime factorization (in particular that of the exponents of the prime factors),

x = 2

and 2x = 1

but these equations do not have a consistent solution. I think the error is in the second reasoning.

Can someone help please?

Cheers

Vivek

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