Given that log two equals 𝑥 and log nine equals 𝑦, find log 72 in terms of 𝑥 and 𝑦.
In order to answer this question, we need to recall our laws of logarithms. However, to start, we need to write the number 72 as multiples of two and nine. 72 is equal to eight multiplied by nine. As two cubed is equal to eight, 72 is equal to two cubed multiplied by nine. We can therefore rewrite log 72 as log of two cubed multiplied by nine.
One of our laws of logarithms states that log 𝑎 plus log 𝑏 is equal to log 𝑎𝑏. This means that log of two cubed multiplied by nine can be rewritten as log two cubed plus log nine. Another one of our laws of logarithms states that 𝑛 log 𝑎 is equal to log 𝑎 to the power of 𝑛. This means that log of two cubed can be rewritten as three log two. Log 72 is equal to three log two plus log nine.
We were told in the question that log two is equal to 𝑥. Therefore, three log two is equal to three 𝑥. Log nine is equal to 𝑦. This means that log 72, written in terms of 𝑥 and 𝑦, is three 𝑥 plus 𝑦.