If \sf a = log_92 and \sf b = log_54, find \sf log_615 in terms of a and b.​

Question

If
`\sf a = log_92` and
`\sf b = log_54`, find
`\sf log_615` in terms of a and b.​

Answer 1

• (4a + b)/(2ab + b)

Explanation:

Given

• a = log9 2
• b = log5 4

To find

• log6 15 in terms of a and b

Solution

Rewrite the given

• log9 2 = a ⇒ log2/log9 = a ⇒ 1/2*log2/log3 = a ⇒ log3 = 1/2*log2/a
• log5 4 = b ⇒ log4/log5 = b ⇒ log5 = 2*log2/b

Rewriting the log6 15:

• log6 15 = log15/log6= (log 5 + log3)/(log2 + log3)

Substitute as follows:

• (log 5 + log3)/(log2 + log3) =
• (2*log2/b + 1/2*log2/a)/(log2 + 1/2*log2/a) =
• (2/b + 1/(2a))/(1 + 1/(2a)) =
• (4a + b)/2ab ÷ (2a + 1)/2a =
• (4a + b)/(2ab + b)