Without using a calculator, find the exact value of: [log3 81 - l...

Question

Without using a calculator, find the exact value of: [log3 81 - log𝝅 1]/[log2√2 8 - log 0.001]

Accepted Answer

determine the exact value of the following expression without the use of a calculator. And you see here we have this log expression, let's first look at log Phase two Pi of one. So what base to pay of what? Let's set this equal to y. Now with the log we can rewrite it in exponential form. If we have the form, log base B of X equals Y. This can be written the next special form as B to the Y equals X. Sophie look here our base is two pi are y is just Y and R. X would be one. So we have two pi to the Y equals one. We want to solve for what well one thing we can tell is that we have two pi to the Y equals one. Any number raised to the power of zero is equal to one. This is one of our principles. Some We have two pi. zero equals 1 and that means Why is equal to zero. So if we were really, Our entire formula will first put zero Not to the same for our next part. Our next part is log base for 64. Well we know this is equal to Y. Our basis for still got a decent red But we have four to the y equals 64. Well, we want to have the base for to determine what our wise So forth. Of the White People's 4 to some number. We should note that four to the third power is equal to 64. If we could multiply four times itself three times we would get 64. This means that our Y. is equal to three. so our next part is plus three and I'll put that in Now it's divided by log of one over 100. Some look at this Log one over equals what? Now it doesn't give us a base. If it doesn't give us a specific base, our base is going to be equal to 10. So our B in this case is 10. You have 10 to the Y equals one over 100. No, you write this tend to the way One over 100 we know is 10 squared. So let's see 10 square. You write this even further into the Y. Now we have a 10 square on the bottom By exponential rules, we have 10 to the -2 or y equals -2. So we get negative two minus our last part. Large Base to split of three of 12th. So people have two squares of three to the way equals 12. So we have to play with this one. Just a little bit to determine how to get here. What I get Base 12 because you want to get rid of that 12 on the right side. Let's go and change up our two squares of three. We know two is actually equal to square root of four multiply that by the spirit of three and then why 12? Now we have square the force and square the three. That is square the 12th. The why equals 12. Now we can go even further. The square root is equal to a number to the power of one half. We get 12 to the power one half. Why equals 12. No you notice here 12 to 1 half Y equals 12. Can I have to imagine there's a one there? In other words we have one half. Y is equal to one because they have the same base, Divide both sides by 1/2 We get y equals two. Sophie substitute this back into our original. You were good minus two at the end. Finally we simplify zero plus three. It's three Divided by -2 -2 is -4. This goes to negative 3/4. We can't simplify any farther. So our answer -3 or four is answer c. Okay. Hope to help you solve the problem. Thank you for watching. Goodbye.

Without using a calculator, find the exact value of: [log3 81 - log𝝅 1]/[log2√2 8 - log 0.001]

Key Concepts

Logarithmic Properties

Change of Base Formula

Exponential and Logarithmic Relationships

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