Math Problem?

1900 (1.08)^z = 800(1.14)^z
divide both sides by 800
19/8 (1.08)^z = (1.14)^z
divide both sides by (1.08)^z

19/8 = (1.14)^z / (1.08)^z
19/8 = (1.14/1.08)^z
log(19/8) = z log(1.14/1.08)

divide both sides by log(1.14/1.08)
log(19/8) / log(1.14/1.08) = z

z = 15.9986

1900 (1 + 0.08)ᶻ = 800 (1 + 0.14)ᶻ
1900 × 1.08ᶻ = 800 × 1.14ᶻ
log(1900 × 1.08ᶻ) = log(800 × 1.14ᶻ)
log(1900) + log(1.08ᶻ) = log(800) + log(1.14ᶻ)
log(1900) + z log(1.08) = log(800) + z log(1.14)
z log(1.08) - z log(1.14) = log(800) - log(1900)
z = [log(800) - log(1900)] / [log(1.08) - log(1.14)]
z ≈ 16.00 (to 4 sig. fig)

1900(1.08)^z = 800(1.14)^z
==> 19/8 = (1.14)^z /(1.08)^z
==> 19/8 = (1.14/1.08)^z
==> log(19/8) = zlog(1.14/1.08)
==> z = (log(19) - log(8))/(log(1.14) - log(1.08))

　
1900 (1 + 0.08)^z = 800 (1 + 0.14)^z
1900/800 = (1 + 0.14)^z/(1 + 0.08)^z
19/8 = (1.14/1.08)^z
19/8 = (19/18)^z
log (19/8) = z log(19/18)
z = log(19/8) / log(19/18)
z = 15.9986

Use the approximation
(1+x)^n, when x is very small as compared with w, then in the first approximation, you can write
(1+x)^n= 1+nx, In this format the above equation becomes,
1900(1+z *0.08) =800(1+z*0.14),
19 + 19*0.08 z= 8+8*0.14z,
1.52 z= -10 +1.12 z ; 0.4 z=10, z= 10/0.4= 25.5. This is an approximate answer.

Math photo b