Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 0.1

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Hello, everybody. I'm doing all right. Today. Today we're gonna be looking at this math question that's asking us to evaluate the following logarithmic expression. And if it is appropriate to write the answer in decimal round it to four decimal places, now, the expression that they provide us is five multiplying the log of 0.1. And the answer choices are a negative five B negative 10 C positive five and D negative 0.2516. Now we're looking at the expression that we have. The first thing I can think of is the decimal 0.1 can be rewritten as a fraction and that would be one over 100. So that's exactly what I'm gonna do now. So we'll have five multiplying the log of one over 100. And why would I want to rewrite it as a fraction? Well, because if I have a fraction with the log, then I can do what's called the deport rule because a fraction is dividing. So let's do a simple recall for that. So the question rule of logarithms, let's, I'll write a general formula for it. Let's say you have the log of base B of A divided by C, that'll be equal to the log of base B of A minus the log of base B of C. So comparing that to the expression that we have the algorithm that we have, we have log with no specified base when this is the case, we can assume, and we use the base 10. So if they don't specify a base, that base is 10, and that 10 will be the B in our recall expression, the numerator or one will be the A in our recall. And the denominator or 100 will be the C in our recall. So using that information, let's rewrite our expression. So we'll still have five outside of everything multiplying log of one minus the log of 100. And now I can think of two other rules that we can have. So we can recall that the log of one is equal to zero. And now there's another rule that I'll, I'll write it first and then I'll explain why we wanna have it. Let's say you have log of base B of being that'll be equal to one. So why is that rule important in this case? Well, I see 100 and I see that we have a base of 10, 100 can be written rewritten as 10 squared. So let's do that. So we'll have five multiplying the log of one minus the log of 10 squared. So as I just mentioned, we have no specified base. So that base is 10. And we're using the recall formula that we have. If you have log B of B, that's equal to one. In this case, we would have log 10 of 10 squared. But we were able to obtain that 10 from the 100. Now to get rid of that square and have it. As we have written it in the recall, we use another rule of the power rule. Let's say I'll write another general formula. Let's say you have log of base B of A to the power C that's equal to C multiplying the log of base B of A. So in this case, you comparing that to the log of 10 squared, our base 10 is the B in the recall, the 10 is the A in the recall. And the second power is the sea in the recall. So let's rewrite this once again, we'll have five multiplying the log of one minus two, multiplying the log of 10. And now we can do some simplification because we're able to get rid of that second power. Now we have log of base 10 of 10 which matches our recall expression of log of base B of B equal to one. So log of 10 will be one. And as we said, also in the recall log of one is zero. So let's rewrite what we have after simplifying. So we have five multiplying the log of one is zero. So zero minus two multiplied by one, which is just two. So we would have five multiplying zero minus two, which is negative two and five multiplying negative two is negative 10, which will be our final answer in this case or let her be. So I hope that was helpful. And until next time.

Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 0.1

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Key Concepts

Logarithms

Common Logarithm

Negative Logarithm Values