Math Snap

STEP 1

1. The annual investment amount $P$ is $5,000.
2. The annual interest rate $r$ is 8%, or 0.08 in decimal form.
3. The desired account balance $A$ is $75,000.
4. We need to find the number of years $n$ it will take for the account to reach $75,000.
5. We will use the given formula for $n$:
$$ n = \frac{\log \left[\frac{A r}{P} + 1\right]}{\log (1+r)}
\]

STEP 2

1. Substitute the given values into the formula.
2. Calculate the intermediate value inside the logarithm.
3. Compute the logarithms.
4. Solve for $n$ and round to the nearest tenth.

STEP 3

Substitute the given values into the formula:
$$n = \frac{\log \left[\frac{75,000 \times 0.08}{5,000} + 1\right]}{\log (1 + 0.08)}$$

STEP 4

Calculate the intermediate value inside the logarithm:
$$\frac{75,000 \times 0.08}{5,000} = \frac{6,000}{5,000} = 1.2$$ So the expression becomes:
$$n = \frac{\log(1.2 + 1)}{\log(1.08)}$$

STEP 5

Compute the logarithms:
$$\log(2.2) \quad \text{and} \quad \log(1.08)$$ Using a calculator:
$$\log(2.2) \approx 0.3424$$ $$\log(1.08) \approx 0.0334$$

Solve for $n$:
$$n = \frac{0.3424}{0.0334} \approx 10.25$$ Round to the nearest tenth:
$$n \approx 10.3$$ The account will be worth $75,000 in approximately $\boxed{10.3}$ years.