Step-by-step explanation:
Sure, let's rearrange the formula \(FV = PV \times (1 + r)^t\) to make \(r\) the subject and then \(t\) the subject.
1. **Make \(r\) the subject:**
\[ FV = PV \times (1 + r)^t \]
Divide both sides by \(PV\):
\[ \frac{FV}{PV} = (1 + r)^t \]
Take the \(t\)th root of both sides:
\[ \left( \frac{FV}{PV} \right)^{\frac{1}{t}} = 1 + r \]
Subtract 1 from both sides:
\[ \left( \frac{FV}{PV} \right)^{\frac{1}{t}} - 1 = r \]
So, \(r\) as a subject is:
\[ r = \left( \frac{FV}{PV} \right)^{\frac{1}{t}} - 1 \]
2. **Make \(t\) the subject:**
\[ FV = PV \times (1 + r)^t \]
Divide both sides by \(PV\):
\[ \frac{FV}{PV} = (1 + r)^t \]
Take the logarithm of both sides:
\[ \log \left( \frac{FV}{PV} \right) = \log \left( (1 + r)^t \right) \]
Apply the power rule:
\[ \log \left( \frac{FV}{PV} \right) = t \cdot \log(1 + r) \]
Divide both sides by \(\log(1 + r)\):
\[ t = \frac{\log \left( \frac{FV}{PV} \right)}{\log(1 + r)} \]
So, \(t\) as a subject is:
\[ t = \frac{\log \left( \frac{FV}{PV} \right)}{\log(1 + r)} \]
