<s>[INST] Logarithmic problem If $\log_{16} 15 =a$ and $\log_{12} 18 =b$, then show that $$\log_{25} 24 = \frac{5-b}{16a-8ab-4b+2}.$$
 [/INST] Let $x=\log _{25}24$ so that $25^x=24$
From $16^a=15$ we get $$2^{8a x}=16^{2a x}=15^{2 x}= 3^{2 x}5^{2x}=3^{2x}24=2^33^{2x+1}$$
$$2^{8a x-3}=3^{2x+1}$$
From $12^b=18$ we get
$$2^{2b-1}=3^{2-b}$$
Multiplying these two equations we have
$$(8a x-3)(2-b)=(2b-1)(2x+1)$$
Solve for $x$
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