Proving Logarithm Equations: 1/log3a + 1/log4a = 1/log12a

[ElementUser]

Homework Statement

Prove:

1/log₃a + 1/log₄a = 1/log₁₂a

Homework Equations

a^y=x
Logarithms rules (addition, subtraction, power, etc.)

log_ax=log_bx/log_ba

The Attempt at a Solution

Left Side:

1/log₃a + 1/log₄a
=log₃a+log₄a/log₁₂a (via common denominator)

The problem is how to add logarithms with different bases. I tried converting the log₃a to log₄a (I get log₄a/log₄3). After that, I subbed it back into the equation.

=log₄a/log₄3+log₄a

But I don't think that gets me anywhere...

Right side still remains the same (1/log₁₂a)

Any help is appreciated! Thanks in advance :).

P.S. What program do people use to make their equations look so neat (the fraction looks real - ex. 1/4 really looks like 1 (horizontal line) 4)?

 

[Dick]

Let's just convert everything to a single log base, like log_12. E.g. log_3(a)=log_12(a)/log_12(3).

 

[ElementUser]

Oh, wow. Sigh, I hate it when you take the wrong approach in proving Left Side equals Right Side.

Thanks for the help! Can't believe it was so simple after your suggestion :).