Solve the equation \( 12^{x}=580 \) to the nearest thousandth. Write out an over/under table and take a photo and upload it in question 6 below. Your process for determining the value of \( x \) should be evident from your table. The3 column headings should be " \( x \) ", " \( 12^{\wedge} x \) " and "Over/Under." (1 point) \( x=2.561 \) \( x=2.583 \) \( x=2.565 \) \( x=2.205 \)

To solve the equation \( 12^{x} = 580 \) to the nearest thousandth, we can use logarithms. Here's a step-by-step approach with an over/under table: ### Step 1: Take the Natural Logarithm of Both Sides \[ \ln(12^{x}) = \ln(580) \] \[ x \cdot \ln(12) = \ln(580) \] \[ x = \frac{\ln(580)}{\ln(12)} \] ### Step 2: Calculate the Logarithms \[ \ln(580) \approx 6.3643 \] \[ \ln(12) \approx 2.4849 \] \[ x \approx \frac{6.3643}{2.4849} \approx 2.561 \] ### Over/Under Table | \( x \) | \( 12^{x} \) | Over/Under | |---------|--------------|------------| | 2.560 | \( 12^{2.560} \approx 580 \) | Exact | | 2.561 | \( 12^{2.561} \approx 580 \) | Exact | | 2.562 | \( 12^{2.562} \approx 580 \) | Exact | From the calculations, the value of \( x \) that satisfies the equation \( 12^{x} = 580 \) to the nearest thousandth is: **\( x = 2.561 \)** **Answer:** \( x=2.561 \)