I am trying to solve the problem of finding the integers x satisfying the inequalities: $2\lt \log_x45\lt3$
I realize this is a very basic question on logarithms and I have the key with the answers 4, 5, 6 and have confirmed this is correct using Wolfram Alpha but which steps should I take to reach that answer?
You can rewrite this as $$2 < \ln(45)/\ln(x) < 3$$ and then $$2\ln(x) < \ln(45) < 3\ln(x)$$ since $\ln(x)$ must be positive. Then this becomes $$\ln(x^2) < \ln(45) < \ln(x^3)$$ and finally exponentiating turns this into the inequality: $$x^2 < 45 < x^3.$$ Now just plug in a few integers to figure out exactly which ones work.