In Exercises 1–40, use properties of logarithms to expand each lo...

Question

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

log(x/100)

Accepted Answer

Hello. Today we're going to be using the properties of logarithms to simplify the given logarithms. So the given logarithms is the logarithms of why? Over 10,000. Now, before we start using our properties, let's just go ahead and rewrite this large number. 10,000 can be rewritten as 10 to the power of four. So rewriting the denominator of our ratio is going to give us the logarithms of why over 10 to the power of four. Okay, so notice how we have a ratio in the arguments of the logarithms. That means that we are dividing why by the number 10 to the power of four. So the argument is evolving division. That means that in order to simplify this given logarithms, we're going to be using the division property of logarithms. And that states that if you have a logo rhythm of a divided by B, you can separate this as a difference of logarithms. Or in other words you can rewrite this as log base A minus log base B. And that's what we're going to be doing here. So A is going to equal to Y. And B is going to be equal to tend to the fourth. So we can rewrite this logo rhythm as the logarithms of why minus the logo rhythm of 10 to the 4th. Okay, now we don't want to have this exponent inside the logarithms. So let's go ahead and rewrite the right hand side using the exponential rule for logarithms. And that states that if you have the logo rhythm of A to the power of B. You can bring this exponent of B in front of the logarithms as a product or in other words the log base A. Of B can be rewritten as B times the logarithms of A. And that's what we're gonna do on the right hand side here. So using our exponential road, we can go ahead and rewrite our difference as the logarithms of why -4 times the logarithms of 10. Now keep in mind that we're working with the general logarithms and the general logarithms has an automatic base of 10. So there is one more simplification that we can do here recall that if you have the log base whose base matches with the argument, then you can go ahead and rewrite this logo rhythm as one. And so here the common logarithms has a base of 10 and the argument is 10. So the base matches with the argument and you can go ahead and rewrite this as one and four times one is just going to give us four. So simplifying this is going to give us the log base of logarithms of why minus four and this is going to be the complete simplification of our given logarithms and that means that the answer to this problem is going to be be. So I hope this video helps you in understanding how to use the properties of logarithms and I'll go ahead and see you all in the next video

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In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log(x/100)