Given that two to the power of 𝑥 equals 18 and 𝑥 is greater than 𝑛 but less than 𝑛 plus one, where 𝑛 is an integer, determine the value of 𝑛.
In order to solve any problem like this, we would normally use the link between logarithms and exponentials. We will look at this method later. However, as we are told that 𝑛 is an integer, we can use our knowledge of powers of two to calculate the answer. We know that two cubed is equal to eight as two multiplied by two multiplied by two is eight. This means that two to the power of four is equal to 16. By multiplying this by two, we can see that two to the power of five is 32. We want two to some power to be equal to 18, and this lies between 16 and 32. This means that 𝑥 lies between the integers four and five. As 𝑛 is the integer less than 𝑥, we can conclude that 𝑛 is equal to four.
As mentioned at the start of the question, we could also solve this problem using logarithms. We know that if 𝑎 to the power of 𝑏 is equal to 𝑐, then 𝑏 is equal to log 𝑐 to the base 𝑎. In this question, we’re told that two to the power of 𝑥 is 18. This means that 𝑥 is equal to log 18 to the base two. Typing this into the calculator gives us 𝑥 is equal to 4.169925 and so on. This value lies between the integers four and five. Once again, we have proved that the value of 𝑛 is four.