To solve the equation \( 3 = \log(\sqrt{w}) - 10w \), we can break it down step by step.

### Step 1: Simplify the logarithmic expression
First, recall that \(\log(\sqrt{w})\) can be rewritten using logarithmic properties. Specifically:
\[
\log(\sqrt{w}) = \log(w^{1/2}) = \frac{1}{2} \log(w)
\]
So the equation becomes:
\[
3 = \frac{1}{2} \log(w) - 10w
\]

### Step 2: Isolate the logarithm
Let's isolate \(\log(w)\) by first adding \(10w\) to both sides:
\[
3 + 10w = \frac{1}{2} \log(w)
\]
Then, multiply both sides by 2 to get rid of the fraction:
\[
6 + 20w = \log(w)
\]

### Step 3: Solve the equation
The equation \( \log(w) = 6 + 20w \) is transcendental and cannot be solved exactly using elementary algebraic methods. However, it can be solved numerically. Here’s how:

1. **Numerical Solution**: You can use numerical methods like Newton-Raphson to approximate the value of \( w \).
2. **Graphical Method**: Alternatively, you can graph the functions \( y = \log(w) \) and \( y = 6 + 20w \) and find the intersection point.

Would you like to proceed with a numerical approximation for \( w \) or graphing the solution?

Learn how to solve the equation 3 = log(sqrt(w)) - 10w step by step with detailed explanations. This problem involves logarithmic properties, transcendental equations, and numerical methods for approximation. Suitable for advanced high school students and above.

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