Question

Use an over/under table to estimate the solution to the exponential equation 3+23x=9
to two decimal places.

Answer 1

To solve the exponential equation \(3 + 23^x = 9\), we first rewrite the equation in a more standard form:

\[
23^x = 9 - 3
\]

This simplifies to:

\[
23^x = 6
\]

Now, we can set up an over/under table to estimate the value of \(x\).

### Step 1: Evaluate \(23^x\) for different values of \(x\)

| \(x\) | \(23^x\) | Location |
|-----------|------------------|------------------|
| 0 | \(23^0 = 1\) | Under (1 < 6) |
| 1 | \(23^1 = 23\) | Over (23 > 6) |
| 0.5 | \(23^{0.5} \approx 4.79\) | Under (4.79 < 6) |
| 0.6 | \(23^{0.6} \approx 7.04\) | Over (7.04 > 6) |
| 0.55 | \(23^{0.55} \approx 5.98\) | Under (5.98 < 6) |
| 0.56 | \(23^{0.56} \approx 6.18\) | Over (6.18 > 6) |

### Step 2: Narrow down the estimation

From our table, we can see that at \(x = 0.55\), \(23^x \approx 5.98\) (which is under 6) and at \(x = 0.56\), \(23^x \approx 6.18\) (which is over 6). Therefore, we know that the solution lies between \(0.55\) and \(0.56\).

### Step 3: Further refine the estimation

We can continue to narrow this down:

| \(x\) | \(23^x\) | Location |
|-----------|------------------|------------------|
| 0.555 | \(23^{0.555} \approx 6.06\) | Over (6.06 > 6) |
| 0.554 | \(23^{0.554} \approx 5.99\) | Under (5.99 < 6) |

### Conclusion

We can see that the value of \(x\) is between \(0.554\) and \(0.555\). To two decimal places, we approximate \(x\) as:

\[
\boxed{0.55}
\]

Therefore, the estimated solution to the equation \(3 + 23^x = 9\) is approximately \(0.55\).