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Let's solve the given equation:

\[
8 \cdot 10^{3x} = 1
\]

### Step 1: Isolate the exponential part
Divide both sides by 8:

\[
10^{3x} = \frac{1}{8}
\]

### Step 2: Take the logarithm of both sides
We will use the common logarithm (\(\log\)) on both sides since the base is 10:

\[
\log(10^{3x}) = \log\left(\frac{1}{8}\right)
\]

### Step 3: Apply logarithmic rules
Use the property \(\log(a^b) = b \cdot \log(a)\) on the left side:

\[
3x \cdot \log(10) = \log\left(\frac{1}{8}\right)
\]

Since \(\log(10) = 1\), the equation simplifies to:

\[
3x = \log\left(\frac{1}{8}\right)
\]

### Step 4: Simplify the logarithm
\(\log\left(\frac{1}{8}\right) = \log(1) - \log(8) = 0 - \log(8) = -\log(8)\).

Now, find \(\log(8)\):

\[
\log(8) = \log(2^3) = 3 \cdot \log(2) \approx 3 \cdot 0.3010 = 0.9030
\]

Thus:

\[
\log\left(\frac{1}{8}\right) = -0.9030
\]

### Step 5: Solve for \(x\)
Now, substitute \(\log\left(\frac{1}{8}\right) = -0.9030\) into the equation:

\[
3x = -0.9030
\]

Divide both sides by 3:

\[
x = \frac{-0.9030}{3} \approx -0.301
\]

### Step 6: Round to the nearest hundredth
The value of \(x\) rounded to the nearest hundredth is:

\[
x \approx -0.30
\]

Thus, the solution is:

\[
\boxed{x \approx -0.30}
\]

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Would you like more details or have any questions?

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Here are 5 related questions to consider:
1. How do logarithmic properties simplify equations?
2. What is the difference between natural and common logarithms?
3. How can you solve exponential equations with different bases?
4. What is the inverse of a logarithmic function?
5. How do you use logarithms to solve for unknown exponents in science?

**Tip:** Always remember that logarithms are the inverse of exponentials, which makes them essential for solving equations with unknown exponents.

Solve for x, rounding to the nearest hundredth. 8 * 10^{3x} = 1

Learn how to solve the exponential equation 8 * 10^{3x} = 1 by isolating the exponential part and using logarithmic properties, with rounding to the nearest hundredth. This guide covers key concepts in exponential equations and logarithmic identities. Suitable for students in Grades 10-12.

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