Answer:
It takes 47.19 years
Step-by-step explanation:
Your amount on the bank is given by the following equation:
[tex]P(t) = P(0)(1+r)^{t}[/tex]
In which P(t) is your current amount, P(0) is the initial amount, r is the rate it changes, and t is the time since the money has been put on the bank.
I put $500 in the bank.
This means that [tex]P(0) = 500[/tex]
I now have $5000 in the bank.
This means that [tex]P(t) = 5000[/tex]
Every year my money increased by 5%
This means that [tex]r = 0.05[/tex]
How long that it takes?
This is t.
[tex]P(t) = P(0)(1+r)^{t}[/tex]
[tex]5000 = 500(1.05)^{t}[/tex]
[tex](1.05)^{t} = \frac{5000}{500}[/tex]
[tex](1.05)^{t} = 10[/tex]
[tex]\log{(1.05)^{t}} = \log{10}[/tex]
We use the following logarithms property:
[tex]\log{a^{t}} = t\log{a}[/tex]
So
[tex]t\log{1.05} = \log{10}[/tex]
[tex]t = \frac{\log{10}}{\log{1.05}}[/tex]
[tex]t = 47.19[/tex]
It takes 47.19 years
