Question

the graph of a logarithmic function below shows compound interest over time with an initial investment of $500 at an interest rate of 7% compounded annually. how many years will it take the... grow to $900?
options:

• approximately 7 years
• approximately 8 years
• approximately 8.5 years

Explanation:

Step1: Recall compound interest formula

The annual compound interest formula is $A = P(1+r)^t$, where:

• $A = 900$ (target amount),
• $P = 500$ (initial investment),
• $r = 0.07$ (annual interest rate),
• $t$ = time in years (unknown).

Step2: Substitute values into formula

$900 = 500(1+0.07)^t$

Step3: Isolate the exponential term

Divide both sides by 500:
$\frac{900}{500} = 1.07^t$
Simplify: $1.8 = 1.07^t$

Step4: Convert to logarithmic form

Take natural log of both sides:
$\ln(1.8) = t\ln(1.07)$

Step5: Solve for t

Rearrange to solve for t:
$t = \frac{\ln(1.8)}{\ln(1.07)}$
Calculate values: $\ln(1.8) \approx 0.5878$, $\ln(1.07) \approx 0.0677$
$t \approx \frac{0.5878}{0.0677} \approx 8.68$

Step6: Match to graph/options

The calculated value ~8.68 is closest to approximately 8.5 years, which aligns with the graph's trend.

Answer:

approximately 8.5 years