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Question
The exponential equation 3x=12
does not have an integer solution. Use an over/under table to find an estimate of the solution to two decimal places.(1 point)
The solution to the exponential equation is x≈
.

Answer 1

To solve the exponential equation \( 3^x = 12 \) using an over/under table, we start by calculating values of \( 3^x \) for different values of \( x \) to find where \( 3^x \) is just below and just above 12.

Here’s a quick table with some values:

| \( x \) | \( 3^x \) |
|---------|----------------|
| 2 | \( 3^2 = 9 \) |
| 3 | \( 3^3 = 27 \) |

From the table, we see that:

- \( 3^2 = 9 \) (which is less than 12)
- \( 3^3 = 27 \) (which is greater than 12)

Since 12 is between 9 and 27, we’ll try some decimal values to narrow it down further:

| \( x \) | \( 3^x \) |
|---------|----------------|
| 2.1 | \( 3^{2.1} \approx 9.11 \) |
| 2.2 | \( 3^{2.2} \approx 9.78 \) |
| 2.3 | \( 3^{2.3} \approx 10.67 \) |
| 2.4 | \( 3^{2.4} \approx 11.63 \) |
| 2.5 | \( 3^{2.5} \approx 12.37 \) |

From this table, we see that:

- At \( x = 2.4 \), \( 3^{2.4} \approx 11.63 \) (still less than 12)
- At \( x = 2.5 \), \( 3^{2.5} \approx 12.37 \) (more than 12)

Since we found that \( 3^x \) is less than 12 when \( x = 2.4 \) and greater than 12 when \( x = 2.5 \), we can narrow our search to between these two values. Now let's try midpoints:

| \( x \) | \( 3^x \) |
|---------|----------------|
| 2.45 | \( 3^{2.45} \approx 12.00 \) |

At \( x = 2.45 \), \( 3^{2.45} \approx 12.00 \), so let's check values around this:

| \( x \) | \( 3^x \) |
|---------|----------------|
| 2.46 | \( 3^{2.46} \approx 12.17 \) |
| 2.44 | \( 3^{2.44} \approx 11.83 \) |

Now we can see that:

- \( 3^{2.44} \) is slightly under 12.
- \( 3^{2.46} \) is slightly over 12.

Thus, the estimated solution is between \( x = 2.44 \) and \( x = 2.46 \). For two decimal places, we find:

The solution to the exponential equation is \( x \approx 2.45 \).