find value of n -0.65 = -(1.023)^-n

1. Decompose the equation:

-0.65 = -(1.023)^{-n}

2. Divide both sides by -1 to simplify:

0.65 = (1.023)^{-n}

3. Take the natural logarithm (ln) of both sides to facilitate solving for \( n \):

\ln(0.65) = \ln((1.023)^{-n})

4. Apply the logarithm power rule:

\ln(0.65) = -n \ln(1.023)

5. Rearrange to isolate \( n \):

n = -\frac{\ln(0.65)}{\ln(1.023)}

6. Calculate the result:

n\approx18.944

Answer: n=18.944