To solve these equations, you can rewrite them using logarithms and then estimate the value of $x$. Let's go through each one step by step.

Part a: $10^x = 18,700$

1. Take the logarithm of both sides: $\log(10^x) = \log(18,700)$

2. Apply the logarithmic rule $\log(10^x) = x \log(10)$ and since $\log(10) = 1$, this simplifies to: $x = \log(18,700)$

3. Use a calculator to find $\log(18,700)$: $x \approx 4.272$

Part b: $10^x = 4,393,000$

1. Take the logarithm of both sides: $\log(10^x) = \log(4,393,000)$

2. Apply the logarithmic rule: $x = \log(4,393,000)$

3. Use a calculator to find $\log(4,393,000)$: $x \approx 6.642$

Final Results:

• For part a, $x \approx 4.272$
• For part b, $x \approx 6.642$

Would you like more details on any of these steps or have any other questions?

Follow-up questions:

1. What is the logarithmic rule for powers of 10?
2. Can you explain why $\log(10) = 1$?
3. How would the process change if the base was not 10 but another number?
4. How do we use natural logarithms in similar problems?
5. Can you describe the general process for solving exponential equations using logarithms?

Tip:

When dealing with large numbers, logarithms simplify calculations and help you quickly find the exponent.