SOLUTION: Approximate the logarithm using the properties of logarithms, given the values logb 2 ≈ 0.3562, logb 3 ≈ 0.5646, and logb 5 ≈ 0.8271. Round your result to four de

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Approximate the logarithm using the properties of logarithms, given the values logb 2 ≈ 0.3562, logb 3 ≈ 0.5646, and logb 5 ≈ 0.8271. Round your result to four de      Log On


   

Question 678074: Approximate the logarithm using the properties of logarithms, given the values logb 2 ≈ 0.3562, logb 3 ≈ 0.5646, and logb 5 ≈ 0.8271. Round your result to four decimal places.
logb 32/125
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The "trick" with these problems is to use properties of logarithms to rewrite the desired log in terms of the logs you've been given. So in this problem we will use properties to transform log%28b%2C+%2832%2F125%29%29 into an expression involving logs of 2, 3 and/or 5.

With a little effort we should be able to figure out that 32+=+2%5E5 and 125+=+5%5E3. So log%28b%2C+%2832%2F125%29%29 can be written as:
log%28b%2C+%282%5E5%2F5%5E3%29%29
Now we can start using properties to rewrite this in terms of logs of 2 and 5. First we use the property log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29 to split the numerator and denominator into separate logs:
log%28b%2C+%282%5E5%29%29+-+log%28b%2C+%285%5E3%29%29
Next we can use the property log%28a%2C+%28p%5En%29%29+=+n%2Alog%28a%2C+%28p%29%29 to move the exponents out of the arguments:
5%2Alog%28b%2C+%282%29%29+-+3%2Alog%28b%2C+%285%29%29
We now have log%28b%2C+%2832%2F125%29%29 expressed in terms of logs of 2 and 5. All we have to do now is substitute in the given values for these logs:
5%2A%280.3562%29+-+3%2A%280.8271%29
Simplifying...
1.781+-+2.4813
-0.7003