Find values of a and b if (21.4) power a=(0.00214) power b=100
Answers
Q -> Find the values of a and b if = 100
solution : (21.4)^a = (0.00214)^b = 100
⇒(21.4)^a = (21.4/10000)^b = 100
⇒(21.4)^a = (21.4)^b/(10000)^b = 100
⇒(21.4)^a = (21.4)^b/(10⁴)^b = 100
⇒(21.4/^a = (21.4)^b/10^4b = 100
case 1 : (21.4)^a = 100
taking log base 10 both sides we get,
⇒a log(21.4) = log(100)
⇒a log(21.4) = 2 .......(1)
case 2 : (21.4)^b/10^4b = 100
⇒(21.4)^b = 100 × 10^4b = 10^2 × 10^4b
⇒(21.4)^b = 10^(2 + 4b)
taking log base 10 both sides we get,
⇒b log(21.4) = log{10^(2 + 4b)}
⇒b log(21.4) = 2 + 4b ........(2)
from equations (1) and (2) we get,
[a log(21.4)]/[b log(21.4) ] = 2/(2 + 4b)
⇒a/b = 2/(2 + 4b)
⇒a/b = 1/(1 + 2b)
⇒a + 2ab = b
⇒2ab = b - a
⇒2 = 1/a - 1/b
Therefore the value of 1/a - 1/b = 2.
To find value of a
from equation (1) , a = 2/log(21.4) ≈ 1.5
To find value of b
from equation (2), b log(21.4) = 2 + 4b
⇒b[log(21.4) - 4] = 2
⇒b = 2/[log(21.4) - 4] ≈ -0.75
Therefore the values of a and b are 1.5 and -0.75 respectively.