If 𝑎 to the power of 𝑦 is equal to 𝑏 to the power of 𝑥 such that 𝑎 and 𝑏 are in the real positive numbers and 𝑎 is not equal to 𝑏, find 𝑑𝑦 by 𝑑𝑥.
In order to find 𝑑𝑦 by 𝑑𝑥, the first thing we’re going to need to do is to get this equation in terms of 𝑦. In our equation, 𝑦 is part of an exponential, as it is in 𝑎 to the power of 𝑦. In order to get the 𝑦 out of the exponential, we’re going to need to perform the inverse operation.
Since we have 𝑎 to the power of 𝑦, we’re going to need to take the log of both sides with a base of 𝑎. So taking the log to base 𝑎 of both sides of this equation gives us log to base 𝑎 of 𝑎 to the power of 𝑦 is equal to log to base 𝑎 of 𝑏 to the power of 𝑥.
Our next step will be to use a logarithm rule. This logarithm rule tells us log to base 𝑘 of 𝑀 to the power of 𝑛 is equal to 𝑛 timesed by the log to base 𝑘 of 𝑀. What this is telling us is that we can take our power and bring it out to the front of the logarithm. And we can do this to 𝑥 also. This gives us that 𝑦 timesed by log of base 𝑎 of 𝑎 is equal to 𝑥 timesed by log of base 𝑎 of 𝑏.
Next, we’ll be using another logarithm rule. This rule tells us that log to base 𝑘 of 𝑘 is equal to one. And so, therefore, log to base 𝑎 of 𝑎 is also equal to one. This gives us that 𝑦 is equal to 𝑥 multiplied by the log of base 𝑎 of 𝑏. Now we have found 𝑦 in terms of 𝑥. And we’re ready to differentiate to find 𝑑𝑦 by 𝑑𝑥.
We simply differentiate this whole equation with respect to 𝑥. And this gives us 𝑑𝑦 by 𝑑𝑥 is equal to log to base 𝑎 of 𝑏. The reason why we get log to base 𝑎 of 𝑏 here is because log to base 𝑎 of 𝑏 is a constant. And 𝑥 simply differentiates to one. So we end up with one timesed by the log to base 𝑎 of 𝑏, leaving us with just the log to base 𝑎 of 𝑏. So this is the solution to the question.