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Laws of logarithms

Dr Lisa Mott discusses the laws of logarithms. Watch the video and use the comments box below to discuss the problem presented at the end of the video
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In this video, we are going to look at the laws of logarithms. So our first law tells us that the logarithm base a of x at the logarithm base a of y is the logarithm base a of x multiplied by y. So we’re going to simplify the logarithm of 2.5 at the logarithm of 4. So using law 1 with x is 2.5 and y is 4, we get that this can be written as the logarithm of 2.5 multiplied by 4. And 2.5 multiplied by 4 is 10 so this is the logarithm of 10. Now we can simplify this even more. First we have to remember if there’s no base given on the logarithm, then the base is actually 10.
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So log of 10 is the same as the logarithm base 10 of 10. And now we’re going to use what we know what a logarithm is to make this simpler. So if x is log base a of b then this is the equivalent of writing the a to the power of x is equal to b. So let’s say the x is log base 10 of 10, so we want to find what x is. Then this is equivalent by having a is equal to 10 and b is equal to 10 of writing 10 to the power of x is equal to 10. So we see that log of 10 is equal to 1, because x must be equal to 1.
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So in our second example, we’re going to use the first law again to simplify the logarithm base y of x at the logarithm base y of b. So using law 1 again, we can see that this would be equivalent, the base would be y and we would just multiply x and b together in the logarithm using Law 1. So this can be simplified as logarithm base y of b x. So I’ve introduced Law 1, now Law 2 tells us that the logarithm base a of x subtract the logarithm base a of y is the same as the logarithm base a of x divided by y.
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So we want to write as a single logarithm, the logarithm base a of 10 subtract the logarithm base a of 8 at the logarithm base a of 2.
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So first of all, we can see if we use Law 2 to simplify the logarithm base a of 10 subtract the logarithm base a of eight then with x is 10 and y is 8, this will be the same as the logarithm base a of 10 over 8. Add still the logarithm base a of 2. The logarithm a of 10/8 can be simplified to the logarithm base a of 5/4. And we’ve still got add the logarithm base a of 2. And using our addition rule, Law 1, we can see that if we add 2 logarithms with the same base, then this is the same as the logarithm with that base of x multiplied by y.
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So we multiply 5/4 and 2 together in the combined logarithm, which simplifies to the logarithm base a 5/2.
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So Law 3 involves powers and it tells us that the logarithm base a of x to the power of k is the same as k multiplied by the logarithm base a of x. So we’re going to solve this equation. So we have some unknown x, and we know logarithm base 3 of x is equal to 2 lots of the logarithm base 3 of 10 subtract logarithm based 3 of 20. Well, we can see we can use Law 3 on 2 times the logarithm base 3 of 10, because in our case k would be 2, a would be 3, and x would be 10. So we can keep the left hand side of the equation the same.
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And we’re first going to write 2 log base 3 of 10 as log base 3 of 10 to the power of k, which is 2 in this case. And we’re still going to have to subtract logarithm base 3 of 20. Now we know 10 squared is 100. And then we can use Law 2 which tells us that if we subtract 2 logarithms with the same base then it is the same as the combined logarithm base a of the first one, x, divided by the second one, y. So in our case, a is 3 and x is 100 and y is 20 to get the logarithm based 3 of x is the logarithm base 3 of 100 divided by 20.
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And 100 divided by 20 is 5. So we’ve solved our equation, because we can see x must be equal to 5, in this case.
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So we’ve gone through the laws of logarithms. In the comments below this video, can you help solve the following. So can we use the law of logarithms
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to solve the following equation: to lots of the logarithm base 3 of x subtract 10 subtract the logarithm base 3 of x plus 1 is equal to 3. So in the comments below this video, can you either A suggest one step for solving this equation or B suggest a common mistake that someone may make when solving this question? Thank you for watching this video.

This video shows Dr Lisa Mott outlining the laws of logarithms.

In the comments below:

Use the law of logarithms to either:
a) Suggest one step for solving the equation shown in the video OR
b) Suggest a common mistake that someone may make when solving this question.

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