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) $$ x=2.561 $$ $$ x=2.583 $$ $$ x=2.565 $$ $$ x=2.205 $$

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UpStudy AI · Accepted Answer

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:** $$ x=2.561 $$
$$ x = 2.561 $$

Solve the equation 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 should be
evident from your table. The3 column headings should be " ", " " and “Over/Under.”
(1 point)

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		Over/Under
2.205	44.66	Under
2.565	586.52	Over
2.561	580.03	Approximates
2.583	592.21	Over

Solve the equation 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 should be evident from your table. The3 column headings should be " ", " " and “Over/Under.” (1 point)