SOLUTION: express logarithm of log base c 5= a and log base c 4 =b find log base c of 2 and log base c of 10

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Question 286328: express logarithm of log base c 5= a and log base c 4 =b
find log base c of 2
and log base c of 10
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28c%2C+%285%29%29+=+a
log%28c%2C+%284%29%29+=+b
To find log%28c%2C+%282%29%29 we will need to use the properties on the base c logarithms we have. There are properties for products, quotients and powers of arguments. Since 4 is a power of 2 we can use log%28c%2C+%284%29%29+=+b and the property about powers in the argument of a logarithm to find log%28c%2C+%282%29%29:
log%28c%2C+%284%29%29+=+b
log%28c%2C+%282%5E2%29%29+=+b
Using the property, log%28x%2C+%28p%5Eq%29%29+=+q%2Alog%28x%2C+%28q%29%29, to move the exponent out of the argument:
2log%28c%2C+%282%29%29+=+b
Now we jsut divide by two:
log%28c%2C+%282%29%29+=+b%2F2

We now have three base c logarithms. For log%28c%2C+%2810%29%29 we look for a product, quotient or power of 2, 4 and/or 5 which makes 10. Obviously 2*5=10 so:
log%28c%2C+%2810%29%29
log%28c%2C+%282%2A5%29%29
Now we can use the property for products in an argument, log%28x%2C+%28p%2Aq%29%29+=+log%28x%2C+%28p%29%29+%2B+log%28x%2C+%28q%29%29, to separate the 2 and the 5:
log%28c%2C+%282%29%29+%2B+log%28c%2C+%285%29%29
We can replace these logarithms with what we know them to be:
b%2F2+%2B+a