### Video Transcript

What is the value of log to the base 0.2 of five?

Okay, if we look at this question, we can see that we have log to base 0.2 of five. Okay, this is actually in logarithmic form. So it’s in the form log to base 𝑎 of 𝑥 equals 𝑏. Now what I’m gonna do is I’m actually gonna turn it into the exponential form to actually help us solve it.

And if we look at the exponential form, we have 𝑎 to the power of 𝑏 is equal to 𝑥. And we can actually use this to solve our problem. Well for our expression, we’ve got log to the base 0.2 of five. I’m gonna say this is equal to 𝑥.

We’ll use 𝑥 as the actual value that we’re looking for. So now if I rewrite this, I can say that actually in the exponential form 0.2 to the power of 𝑥 is equal to five. And it’s now 𝑥 that we trying to find because again 𝑥 is the value of log to the base 0.2 of five.

In order to do that and actually solve this, what we’re gonna do is we’re actually gonna have a look at changing 0.2 into a fraction. And when we do that, we can see this is very useful actually because we get one-fifth to the power of 𝑥 is equal to five.

The key thing in this question is that actually one-fifth is the reciprocal of five. And this is a relationship that’s really useful because it means that we now know that 𝑥 must be equal to negative one.

Now we know that because we have the rule that 𝑥 to the power of negative 𝑎 is equal to one over 𝑥 to the power of 𝑎. So great! We’ve solved the problem. We could say the value of log to the base 0.2 of five is equal to negative one.