To find the amount of time $t$ after which the employee will earn $10.00 per hour, we start with the equation:

$y = 7.89 \times (1.04)^t$

We want to set $y = 10.00$:

$10.00 = 7.89 \times (1.04)^t$

Next, we can isolate $(1.04)^t$:

$(1.04)^t = \frac{10.00}{7.89}$

Calculating the right side:

$\frac{10.00}{7.89} \approx 1.2673$

Now we have:

$(1.04)^t = 1.2673$

To solve for $t$, we take the logarithm of both sides:

$\log((1.04)^t) = \log(1.2673)$

Using the property of logarithms:

$t \cdot \log(1.04) = \log(1.2673)$

Now, solve for $t$:

$t = \frac{\log(1.2673)}{\log(1.04)}$

Calculating the logarithms:

$t \approx \frac{0.1031}{0.0170} \approx 6.06$

Thus, the employee will earn $10.00 per hour after approximately 6.06 years.

Summary

The employee will reach an hourly wage of $10.00 after about 6.06 years.

Would you like more details on any part of this process? Here are some related questions you might consider:

1. How would the calculation change with a different starting hourly wage?
2. What would the total earnings be over that time period at the new wage?
3. How would the time to reach $10.00 differ with a different raise percentage?
4. What is the cumulative effect of raises over a longer period, say 10 years?
5. How can we model future raises if they are not constant?

Tip: When solving exponential equations, logarithms are often your best tool for isolating the variable.