Solving Exponential Equations Grade 11 Math?

a) plug 10 in for n and use a caculator:
P(10) = 12(1.025^10)

b) solve for n using logarithms:

24 = 12(1.025^n)
2 = 1.025^n
n = log 2 / log 1.025

c) solve for n using logarthims:

12 = .8(1.025^n)

end fixing for x 3^(3x+a million) = 3/27 take logs (3x+a million)log(3) = log(a million/9) (3x+a million)log3 = log1 - log9 (3x+a million)log3 = 0 - log(3^2) (3x+a million)log3 = -2log3 (3x+a million) = -2log3/log3 3x+a million = -2 3x = -3 x= -a million does this make it seem simplier?

I dont have calculator with me, please pluck in the expression in the final answer. SOrry.

a.
subsitute n=10,
P(10) = 12(1.025^10)

b.
the initial population is 12000 , the double of that is 24000, but notice the expression for P(n) is in thousands. Hence P(n) is actually just 24
therefore, 24 = 12(1.025^n)
2 = 1.025^n
ln(2) = ln(1.025^n)= n* ln(1.025)
n = ln2 / ln1.025 which is positivive

c.
population now is 800, and require to find n
once again P(n) is in thousands hence P(n) = 0.8
0.8 = 12(1.025^n)
0.8/12 = 1.025^n
n = [ln (0.8/12)] / ln1.025
this answer is negative , hence the answer would be so so many years ago

a) let n = 10
P(10) = 12(1.025)^10 = 15.361 = 15,361 people

b) population double will be 24000

24= 12(1.025)^n divide by 12
2 = (1.025)^n

take the log of both sides

log2 = log(1.025)^n
log2 = nlog(1.025)
log2/log(1.025) = n
28.07 = n

about 28 years to double

c) 800 in terms of thousands = 800/1000 = 0.8

0.8 = 12(1.025)^n
divide b 12

1/15 = (1.025)^n
take the log

log(1/15) = n log (1.025)
log(1/15)/log(1.025) = n
-109.7

about 110 years ago

P(0) = 12000 {Current population}

P(n) = population, in thousands, in n years

a) We need P(10) = 12 (1.025^10) * 1000

b) We need to find when P(n) = 24
12 (1.025^n) = 24
1.025 ^n = 2
n = log (base 1.025) 2

n = ln 2 / ln 1.025

ln 2 = approx. 0.69315
ln 1.025 = approx 0.02469
n = about 28 years.

c) P(n) = 800
12000 * 1.025^n = 800
1.025 ^n = 800/12000
n = ln (800/12000) / ln 1.025
n = approx. -110
The population was 800 about 110 years ago.