If log _{30} 3=a and log _{30} 5=b, then find | vedclass

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Question

(A) Given that $\log _{30} 3 = a$ and $\log _{30} 5 = b.$We know that $\log _{30} 15 = \log _{30} (3 	imes 5) = \log _{30} 3 + \log _{30} 5 = a + b.$

Vedclass · Accepted Answer

$3(1-a-b)$

Vedclass · Answer

(A) Given that $\log _{30} 3 = a$ and $\log _{30} 5 = b.$We know that $\log _{30} 15 = \log _{30} (3 	imes 5) = \log _{30} 3 + \log _{30} 5 = a + b.$Also,$\log _{30} 15 = \log _{30} \left(\frac{30}{2}ight) = \log _{30} 30 - \log _{30} 2 = 1 - \log _{30} 2.$Equating the two expressions for $\log _{30} 15$:$a + b = 1 - \log _{30} 2 \Rightarrow \log _{30} 2 = 1 - a - b.$Now,$\log _{30} 8 = \log _{30} (2^3) = 3 \log _{30} 2.$Substituting the value of $\log _{30} 2$:$\log _{30} 8 = 3(1 - a - b).$

If $\log _{30} 3=a$ and $\log _{30} 5=b,$ then find the value of $\log _{30} 8.$

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