Question

Use an over/under table to estimate the solution to the exponential equation 3+23x=9
to two decimal places.(1 point)
The solution to the exponential equation is x≈
.

Answer 1

To solve the exponential equation \( 3 + 23^x = 9 \), let's first isolate \( 23^x \):

\[
23^x = 9 - 3
\]
\[
23^x = 6
\]

Next, we can take the logarithm of both sides to solve for \( x \):

\[
x = \log_{23}(6)
\]

Using the change of base formula:

\[
x = \frac{\log(6)}{\log(23)}
\]

Now we can compute this using logarithm values. Calculating these using a calculator:

\[
\log(6) \approx 0.7781
\]
\[
\log(23) \approx 1.3617
\]

Now to divide these:

\[
x \approx \frac{0.7781}{1.3617} \approx 0.571
\]

Now, we need to confirm this using an over/under table. We'll check values around \( x \approx 0.57 \):

### Estimate Table

| \( x \) | \( 23^x \) | \( 3 + 23^x \) | Comparison with 9 |
|---------|------------|----------------|-------------------|
| 0.50 | \( 23^{0.50} \approx 4.796 \) | \( 3 + 4.796 \approx 7.796 \) | Under |
| 0.60 | \( 23^{0.60} \approx 6.616 \) | \( 3 + 6.616 \approx 9.616 \) | Over |

From the table:
- For \( x = 0.50 \), \( 3 + 23^{0.50} \) gives approximately 7.80 (which is under 9).
- For \( x = 0.60 \), \( 3 + 23^{0.60} \) gives approximately 9.62 (which is over 9).

This indicates that the solution \( x \) is between 0.50 and 0.60. To narrow it down further, we can try an average of these two:

### Narrowing Down Further

1. **Try \( x = 0.55 \)**:

\[
23^{0.55} \approx 5.696 \rightarrow 3 + 5.696 \approx 8.696 \quad (\text{Under})
\]

2. **Try \( x = 0.57 \)**:

\[
23^{0.57} \approx 6.049 \rightarrow 3 + 6.049 \approx 9.049 \quad (\text{Over})
\]

So we can conclude that:

\[
x \text{ is between } 0.55 \text{ and } 0.57.
\]

3. **Try \( x = 0.56 \)**:

\[
23^{0.56} \approx 5.866 \rightarrow 3 + 5.866 \approx 8.866 \quad (\text{Under})
\]

Now we can establish:

- \( 0.56 \) is under 9.
- \( 0.57 \) is over 9.

Using interpolation, we can conclude that a more precise estimate is:

- The solution to the exponential equation is approximately \( x \approx 0.57 \).

### Final Answer
Thus, the final answer is:

\[
x \approx 0.57.
\]