Answer:
log 0.045=1-log 2 -2 - log (small)e 11/10
or
-1.346
Explanation:
log0.045=log 9/200
We can use the property of logarithms that states:
log(small)b a/c = log (small)b a - log (small)b c
applying this property, we get:
log 9/200 = log 9 - log 200
simplify:
log 200=log 2+ log 100=log 2+2
substitute this back into the original equation:
log 0.045 = log 9 - log 200 = log 9 - (log 2+2)
Use the fact that log 10=1 to simplify log 9:
log 9=log(10-1)=log 10 +log (1-1/10)=1-log 10 ^-1 + Reiman's sum (from n=1 to infinity) 1/n (1/10)^n
Since log 10=1, we have log 10^-1=-1, so we get:
log 9 = 1+1 - Reiman's sum (from n=1 to infinity) 1/n (1/10)^n
Substituting back into the original equation we get:
log 0.045=(1+1- Reiman's sum (from n=1 to infinity) 1/n (1/10)^n)-(log 2+2)
This is a convergent series that sums to:
log 0.045=1-log 2 -2 - log (small)e 11/10
Simplifying this expression we get:
log 0.045 = -1.346
You would probably give log 0.045=1-log 2 -2 - log (small)e 11/10 if you're not allowed to use a calculator.