math question on logs?

Question

how would you solve: 9(1.03)^t=11(1.02)^t

Answer 1

(1.03/1.02)^t =11/9 so
t = log(11/9)/log(1.03/1.02) = 20.5686

Answer 2

9(1.03)^t=11(1.02)^t

divde 11 for both sides
(9/11) (1.03)^t = 1.02^t

divide (1.03)^t for both sides
9/11 = (1.02)^t / (1.03)^t

we know that a^x / b^x = (a/b)^x

so
9/11 = (1.02/1.03)^t

take log for both sdies
log(9/11) = log (1.02/1.03)^t

log rule:
log(a^x) = x log(a)

log(9/11) =t log (1.02/1.03)

divide log (1.02/1.03) for both sides
t = log(9/11) / log (1.02/1.03) <== answer

now grab a calculator and you'll get t=~ 20.869


hope it hels
Rec

Answer 3

9(1.03)^t=11(1.02)^t
1.03^t=11/9(1.02)^t
1.03^t=1.2222(1.02)^t
1.03^t/1.02^t=1.2222
(1.03/1.02)^t=1.2222
1.0098^t=1.2222
taking log both sides
t*log1.0098=log 1.2222
t=log1.2222/log1.0098
using calculator t=20.57

Answer 4

9(1.03)^t=11(1.02)^t

Take the ln of both sides
ln (9(1.03)^t )= ln(11(1.02)^t )

ln9 +t ln(1.03) = ln(11) +t ln(1.02)
t ( ln(1.03 -ln(1.02) ) = ln(11) - ln(9)
t = ( ln(11) - ln(9) ) / ( ln(1.03) -ln(1.02) )

Now simplify or evaluate it