how would you use log which has a subscript of 2 5 to approximate log subscript 2 50 and log subscript 2 0.4?
log(2) 50 = log(2) [5^2*2] =2 log(2)5 + log(2) 2
log(2) 0.4 = log(2) (2/5) = log(2) 2 - log(2) 5
But log(2) 2 = 1, so
log(2) 50 = log(2) [5^2*2] = 2 log(2)5 + 1
and
log(2) 0.4 = log(2) (2/5) = 1 - log(2) 5
To approximate logarithms with a different base, such as log₂50 and log₂0.4, you can use the Change of Base Formula. This formula allows you to convert a logarithm with one base to a logarithm with another base. Here's how you can use log₂5 to approximate log₂50 and log₂0.4:
1. Change the base of the logarithm you want to approximate. You can use either the natural logarithm (ln) or logarithm base 10 (log) for this step. Let's use ln.
log₂50 ≈ ln50 / ln2
log₂0.4 ≈ ln0.4 / ln2
2. Use a calculator or a math software to evaluate the logarithms on the right side of the formulas.
log₂50 ≈ 3.3219 / 0.6931
log₂0.4 ≈ -1.3219 / 0.6931
3. Simplify the expressions:
log₂50 ≈ 4.7813
log₂0.4 ≈ -1.9056
So, using log₂5 to approximate log₂50, you would get approximately 4.7813, and to approximate log₂0.4, you would get approximately -1.9056.