Express as a simple logarithm 2ln6 - ln3 .
We start with: 2ln6 - ln3 ... First, we rewrite this expression as: ln6 + ln6 - ln3 ... Next, we rewrite this as: ln(23) + ln(23) - ln3 ... Using the log rule logaxy = logax + logxy, we express this as ln2 + ln3 + ln2 + ln3 - ln3 ... We simplify this to ln2 + ln2 + ln3 ... Using the log rule logax + logay = logaxy, we express this as ln (223) ... Finally, we can simplify this to ln12. Alternative method: We start with: 2ln6 - ln3 ... First, using the log rule: ylogax = loga(xy) we express this as ln(62)- ln3 ... Next, we rewrite this as: ln36 - ln3 ... Using the log rule logax - logxy = loga(x/y) we express this as ln(36/3) ... Finally, we can simplify this to ln12
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