Answer:
[tex]6000 = 3000(1+0.0424)^{t}[/tex]
t = 16.73 years
Step-by-step explanation:
This can be solved using the "exponential growth" formula.
Steps:
A = Final Amount
S = Starting Value
r = rate
c = times in a year ( c = 1 in this equation which is why it's not shown in the actual equation )
1. [tex]A=S(1+\frac{r}{c})^{ct}[/tex] → cancel out S → [tex]\frac{A}{S} = \frac{S(1+\frac{r}{c})^{ct} }{S}[/tex]
2. [tex]\frac{A}{S} = {(1+\frac{r}{c})^{ct} }[/tex] → sperate ct from equation by logging both sides (logging a value with an exponent brings the exponent in front of log) → [tex]log(\frac{A}{S}) = ctlog(1+\frac{r}{c})[/tex]
3. [tex]log(\frac{A}{S}) = ctlog(1+\frac{r}{c})[/tex] → transfer right side log to left side along with the c value to only have t remaining → [tex]\frac{log(\frac{A}{S})}{clog(1+\frac{r}{c})} = \frac{ctlog(1+\frac{r}{c})}{clog(1+\frac{r}{c})}[/tex]
4. Solve for answer: [tex]\frac{log(\frac{A}{S})}{clog(1+\frac{r}{c})} = t[/tex]
