Solve the equation $$ 12^{x}=580 $$ to the nearest thousandth, Write out an overfunder 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 " $$ x $$ " and "Overfunder," (1 point) c) $$ x=2.561 $$ $$ x=2.583 $$ $$ x=2.565 $$ $$ x=2.205 $$

Question

UpStudy AI · Accepted Answer

To solve the equation $$ 12^{x} = 580 $$ to the nearest thousandth, we'll follow these steps:

### Step 1: Take the Natural Logarithm of Both Sides
$$
\ln(12^{x}) = \ln(580)
$$
$$
x \cdot \ln(12) = \ln(580)
$$

### Step 2: Solve for $$ x $$
$$
x = \frac{\ln(580)}{\ln(12)}
$$

### Step 3: Calculate the Values
$$
\ln(580) \approx 6.3662
$$
$$
\ln(12) \approx 2.4849
$$
$$
x \approx \frac{6.3662}{2.4849} \approx 2.561
$$

### Conclusion
The value of $$ x $$ to the nearest thousandth is **$$ x = 2.561 $$**.

### Overfunder Table

Below is an **Overfunder** table that helps illustrate the process of determining the value of $$ x $$:

| $$ x $$ | $$ 12^{x} $$ | Overfunder          |
|---------|--------------|---------------------|
| 2.56    | 12².⁵⁶ ≈ 579.26 | Under (579.26 < 580) |
| **2.561** | **12².⁵⁶¹ ≈ 580.00** | Exact Approximation |
| 2.57    | 12².⁵⁷ ≈ 585.75 | Over (585.75 > 580) |

*Note: The "Overfunder" column indicates whether the calculated $$ 12^{x} $$ is an overestimation or underestimation compared to 580.*

### Answer Choice

Based on the calculations and the table above, the correct answer is:

**c) $$ x = 2.561 $$**
The value of $$ x $$ is approximately **2.561**.

		Overfunder
2.205	76.12	Underfunded
2.565	580.01	Funded
2.583	598.18	Overfunded

Solve the equation to the nearest thousandth, Write out an overfunder 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 " ", " 12 " " and “Overfunder,”
(1 point)
c)

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Solve the equation to the nearest thousandth, Write out an overfunder 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 " ", " 12 " " and “Overfunder,” (1 point) c)