Find the solution set of log base 𝑥 of five equals log base three of three for all real values.
In order to answer this question, we will recall some of our laws of logarithms. Firstly, we know that log base 𝑎 of 𝑎 is equal to one. This means that the right-hand side of our equation log base three of three is also equal to one. We can therefore rewrite the equation as log base 𝑥 of five equals one.
It might be obvious at this stage that 𝑥 must be equal to five. One way of checking this would be to recall that if log base 𝑥 of 𝑎 is equal to 𝑏, then 𝑥 to the power of 𝑏 is equal to 𝑎. This means that we can rewrite the equation log base 𝑥 of five equals one as 𝑥 to the power of one equals five. Since any real number raised to the power of one is itself, then 𝑥 equals five. The solution set to the equation log base 𝑥 of five equals log base three of three is five.