What is the value of log to the base 0.04 of 0.2 to the power of negative three?
Well, the first thing we can do with our expression is convert our base. And we can convert our base into 0.2 squared. And that’s because 0.04 is equal to 0.2 multiplied by 0.2. So when we do that, what we have is log to the base 0.2 squared of 0.2 to the power of negative three. But you might think, “Well, why have we done that?” Well, we’ve done that because of one of our log laws. And that log law states that log to the base 𝑎 to the power of 𝑚 of 𝑎 to the power of 𝑛 is equal to 𝑛 over 𝑚, and this only occurs where 𝑚 is not equal to zero.
So, if we take a look at our expression, the first thing we need is for our 𝑎-value to be the same. And we can see it is because in the base, we have 0.2, and then we also have 0.2 here on the right-hand side. And then, we can see that our 𝑚-value would be two because that’s what the 0.2 is raised to when we’re looking at the base. And then, we also have our 𝑛-value, which is negative three. So therefore, what we can say is this is gonna be equal to negative three over two, and we’ve got that by using our log law.
So, we can say that the value of log to the base 0.04 of 0.2 to the power of negative three is negative three over two.