To solve the equation \( 770 = 610 (1.06)^{0.68x + 2} - 150 \), follow these steps:

1. **Isolate the exponential term**:
   \[
   770 + 150 = 610 (1.06)^{0.68x + 2}
   \]
   \[
   920 = 610 (1.06)^{0.68x + 2}
   \]

2. **Divide both sides by 610**:
   \[
   \frac{920}{610} = (1.06)^{0.68x + 2}
   \]
   \[
   1.5082 \approx (1.06)^{0.68x + 2}
   \]

3. **Apply the natural logarithm to both sides**:
   \[
   \ln(1.5082) = \ln\left( (1.06)^{0.68x + 2} \right)
   \]

4. **Use the power rule of logarithms** \( \ln(a^b) = b \ln(a) \):
   \[
   \ln(1.5082) = (0.68x + 2) \ln(1.06)
   \]

5. **Solve for \( 0.68x + 2 \)**:
   \[
   0.68x + 2 = \frac{\ln(1.5082)}{\ln(1.06)}
   \]

6. **Calculate the logarithms**:
   \[
   \ln(1.5082) \approx 0.410
   \]
   \[
   \ln(1.06) \approx 0.0583
   \]

7. **Substitute and solve**:
   \[
   0.68x + 2 = \frac{0.410}{0.0583} \approx 7.033
   \]

8. **Isolate \( x \)**:
   \[
   0.68x = 7.033 - 2
   \]
   \[
   0.68x = 5.033
   \]
   \[
   x = \frac{5.033}{0.68} \approx 7.401
   \]

So, the value of \( x \) is approximately \( 7.401 \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would you solve \( 820 = 500 (1.08)^{0.75x + 1} - 200 \)?
2. What is the effect of changing the base of the exponential function in the equation?
3. How can you solve \( 700 = 650 (1.04)^{0.5x + 3} - 100 \) instead?
4. What are the properties of natural logarithms that are useful in solving exponential equations?
5. How can you verify the solution of an exponential equation involving additional constants?

**Tip:** When solving exponential equations, isolating the exponential term as the first step can simplify the process significantly.

Learn how to solve the exponential equation 770 = 610 (1.06)^(0.68x + 2) - 150 with a detailed step-by-step solution. This problem involves exponential equations and logarithms, covering key mathematical concepts and formulas. Suitable for students in Grades 10-12.

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