Questions: Consider the equation (4 cdot 10^-3 x=18). Solve the equation for (x). Express the solution as a logarithm in base-10. Approximate the value of (x). Round your answer to the nearest thousandth. (x approx)

Solution Steps

To solve the equation \(4 \cdot 10^{-3x} = 18\) for \(x\), we can follow these steps:

1. Isolate the Exponential Term: Divide both sides of the equation by 4 to isolate the term with the exponent.
2. Apply Logarithms: Take the logarithm of both sides to solve for the exponent. Since the base of the exponent is 10, use the base-10 logarithm.
3. Solve for \(x\): Rearrange the equation to solve for \(x\).

Starting with the equation: \[ 4 \cdot 10^{-3x} = 18 \] we divide both sides by 4: \[ 10^{-3x} = \frac{18}{4} = 4.5 \]

Next, we take the base-10 logarithm of both sides: \[ \log_{10}(10^{-3x}) = \log_{10}(4.5) \] Using the property of logarithms, we simplify the left side: \[ -3x = \log_{10}(4.5) \]

Now, we solve for \(x\) by dividing both sides by -3: \[ x = -\frac{\log_{10}(4.5)}{3} \] Calculating the logarithm: \[ \log_{10}(4.5) \approx 0.6532 \] Thus, \[ x \approx -\frac{0.6532}{3} \approx -0.2177 \] Rounding to the nearest thousandth gives: \[ x \approx -0.218 \]

Final Answer