M(x) = log10(x / 0.001)

7.8 = log10(x / 0.001)

10^7.8 = x / 0.001

x = 10^7.8 * 0.001

x =

Let's solve for \( x \) step by step:

Given the equation:
\[ M(x) = \log_{10}\left(\frac{x}{0.001}\right) \]

We have:
\[ 7.8 = \log_{10}\left(\frac{x}{0.001}\right) \]

Convert the logarithmic equation to an exponential equation:
\[ 10^{7.8} = \frac{x}{0.001} \]

Rewriting the right-hand side:
\[ 10^{7.8} = x \cdot 1000 \]

Solve for \( x \):
\[ x = \frac{10^{7.8}}{1000} \]

Calculate \( 10^{7.8} \):
\[ 10^{7.8} \approx 6.30957344480193 \times 10^7 \]

Now divide by 1000:
\[ x = \frac{6.30957344480193 \times 10^7}{1000} \]
\[ x = 6.30957344480193 \times 10^4 \]
\[ x \approx 63095.7344480193 \]

So, the value of \( x \) is approximately:
\[ x \approx 63095.73 \]

Would you like more details or have any questions about the solution?

Here are 5 more questions you can ask:
1. How do you solve logarithmic equations with different arguments?
2. What is the change of base formula for logarithms?
3. How do logarithmic and exponential functions relate to each other?
4. What are some common mistakes to avoid when working with logarithms?
5. How do you solve logarithmic inequalities?

**Tip:** When working with logarithms, remember that \( \log_b(a) = c \) implies \( b^c = a \). This conversion is useful for solving equations.

Learn how to solve the logarithmic equation log10(x / 0.001) = 7.8 with a detailed step-by-step solution. This problem covers logarithms and exponential equations, demonstrating the conversion method and calculations involved. Suitable for high school students.

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