Suppose that a new employee starts working at ​$7.89 per hour and receives a 4​% raise each year. After time​ t, in​ years, his hourly wage is given by the equation y equals $ 7.89 left parenthesis 1.04 right parenthesis Superscript t Baseline . Find the amount of time after which he will be earning​ $10.00 per hour.

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.

Suppose that a new employee starts working at ​$7.89 per hour and receives a 4​% raise each year. After time​ t, in​ years, his hourly wage is given by the equation y equals $ 7.89 left parenthesis 1.04 right parenthesis Superscript t Baseline. Find the amount of time after which he will be earning​ $10.00 per hour.

Discover how long it takes for an employee starting at $7.89 per hour with a 4% annual raise to reach a wage of $10.00 per hour. This detailed solution involves exponential growth and logarithmic calculations, ideal for students in Grades 9-12.

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