Ln(x) = (1/q).Ln(50) → you divide by Ln(5) both sides
Ln(x) / Ln(5) = (1/q).Ln(50) / Ln(5)
Log[5](x) = (1/q).Ln(50) / Ln(5) → you know that: Ln(50) = Ln(25) + Ln(2) = 2.Ln(5) + Ln(2)
Log[5](x) = (1/q).[2.Ln(5) + Ln(2)] / Ln(5)
Log[5](x) = (1/q).[2 + {Ln(2) / Ln(5)}] → recall (3): Ln(2) / Ln(5) = (2p - q)/(3q - p)
Log[5](x) = (1/q).[2 + {(2p - q)/(3q - p)}]
Log[5](x) = (1/q).[{2.(3q - p) + (2p - q)}/(3q - p)]
Log[5](x) = (1/q).[{6q - 2p + 2p - q}/(3q - p)]
Log[5](x) = (1/q).[5q/(3q - p)]
Log[5](x) = 5/(3q - p)
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cidyah say: logx 40 =p
log(40) /log(x) = p
log(40) = p log(x)
log(x) = log(40) /p
logx 50 = q
log(50)/log(x) = q
log(50) = q log(x)
log(x) = log(50)/q
log2 x = log(x) /log(2)
= log(40) / p log(2)
= log(40) / log(2^p)
log2 x = log(40) /log(2^p) ------(1)
log5 x = log(x) / log(5)
= log(50) / q log(5)
= log(50) /log(5^q)
log5 x = log(50) /log(5^q) ------(2)
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Jan say: lg_x(40)= lg(40)/lg(x) => lg(x)=lg(40)/p
and
lg_x(50)= lg(50)/lg(x) => lg(x)=lg(50)/q
so
lg_2(x) = (lg(40)/lg(2)*p)
and
lg_5(x) = (lg(50)/lg(5)*q)
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Captain Matticus, LandPiratesInc say: log[x](40) = p
log[x](50) = q
log[x](8 * 5) = p
log[x](10 * 5) = q
log[x](8) + log[x](5) = p
log[x](10) + log[x](5) = q
3 * log[x](2) + log[x](5) = p
log[x](2) + 2 * log[x](5) = q
3 * log[x](2) + log[x](5) - 3 * (log[x](2) + 2 * log[x](5)) = p - 3 * q
3 * log[x](2) + log[x](5) - 3 * log[x](2) - 6 * log[x](5) = p - 3q
-5 * log[x}(5) = p - 3q
log[x](5) = 3q - p
3 * log[x](2) + log[x](5) = p
3 * log[x](2) + 3q - p = p
3 * log[x](2) = 2p - 3q
log[x](2) = (1/3) * (2p - 3q)
log[2](x) =>
1 / log[x](2) =>
1 / ((1/3) * (2p - 3q)) =>
3 / (2p - 3q)
log[5](x) =>
1 / log[x](5) =>
1 / (3q - p)
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