Logarithm

If$ log_{12} 18=\alpha $[log 18 to the base 12] and $ log_{24} 54=\beta $[log 54 to the base 24] , Prove that $ \alpha\beta+5(\alpha-\beta)=1 $

Please show your steps.

$$log_2{3}=t$$
$$a=\frac{2t+1}{t+2}$$
$$b=\frac{3t+1}{t+3}$$
$$\frac{2t+1}{t+2}\frac{3t+1}{t+3}+5(\frac{2t+1}{t+2}-\frac{3t+1}{t+3})=1$$

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