SOLUTION: How do I set up the problem for the square root of log 425000 without a calculator? I must show my work. Thank you, -Adeline Border

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: How do I set up the problem for the square root of log 425000 without a calculator? I must show my work. Thank you, -Adeline Border       Log On


   

Question 725430: How do I set up the problem for the square root of log 425000 without a calculator? I must show my work. Thank you,
-Adeline Border


Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
It would help if you would include the instructions for the problem. I'm guessing that it says something like "Simplify". If I'm wrong you'll have to re-post your equation (with the instructions).

sqrt%28log%28%28425000%29%29%29
The only way to simplify a logarithm without a calculator (or a table of logarithms) is if the argument is a power of the base or if there is product, quotient or power that involves a power of the base. Since the base of "log" is 10, we are looking for powers of 10.

425000 is not itself a power of 10. (Positive powers of 10 are a 1 followed by zeros, negative powers of 10 are the reciprocals of the positive powers and a zero power of 10 is 1.) But 425000 is a product that involves a power of 10:
425000 = 425 * 1000
So we can rewrite the argument as:
sqrt%28log%28%28425%2A1000%29%29%29
Then we can use a property of logarithms, log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29, to split the log in two (which separates the 425 and 1000):
sqrt%28log%28%28425%29%29%2Blog%28%281000%29%29%29
Since 1000+=+10%5E3 the second log is a 3:
sqrt%28log%28%28425%29%29%2B3%29
This is as far as we can go without a calculator. If you have a table of logarithms and you are allowed to use it, we can go a little farther. We can factor out a power of 10 from 425 so that we get a number that is in the table. Not knowing what you table looks like but it should have a place for 4.25 or for 0.425. I'm guessing it has 4.25 In this case we would factor out 100:
sqrt%28log%28%28100%2A4.25%29%29%2B3%29
Splitting the log again we get:
sqrt%28log%28%28100%29%29%2Blog%28%284.25%29%29%2B3%29
The first log is 2 since 100+=+10%5E2. We can get a decimal approximation for the second log from the table:
sqrt%282%2B0.6284%2B3%29
Then we can add inside the square root:
sqrt%285.6284%29
This is far as we can go using a table of logarithms but not a calculator. (And it is not exactly correct because the table can only provide approximate values for most logarithms.