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 equation \( 3 + 23^x = 9 \), we first isolate the term with \( x \):

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

Next, we apply logarithms to both sides of the equation:

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

To estimate \( x \) using an over/under table, we can try different values of \( x \) and calculate \( 23^x \) until we find values that bracket \( 6 \).

### Over/Under Table

| Value of \( x \) | \( 23^x \) | Over/Under |
|-------------------|--------------|-------------|
| 0 | \( 23^0 = 1 \) | Under |
| 1 | \( 23^1 = 23 \) | Over |
| 0.5 | \( 23^{0.5} \approx 4.79 \) | Under |
| 0.6 | \( 23^{0.6} \approx 6.12 \) | Over |
| 0.55 | \( 23^{0.55} \approx 5.52 \) | Under |
| 0.58 | \( 23^{0.58} \approx 5.93 \) | Under |
| 0.59 | \( 23^{0.59} \approx 6.03 \) | Over |

From the table:

- At \( x = 0.55 \), \( 23^{0.55} \approx 5.52 \) (Under)
- At \( x = 0.58 \), \( 23^{0.58} \approx 5.93 \) (Under)
- At \( x = 0.59 \), \( 23^{0.59} \approx 6.03 \) (Over)

Now we can narrow down further between \( 0.58 \) and \( 0.59 \):

| Value of \( x \) | \( 23^x \) | Over/Under |
|-------------------|--------------|-------------|
| 0.585 | \( 23^{0.585} \approx 5.98 \) | Under |
| 0.588 | \( 23^{0.588} \approx 6.00 \) | Over |

### Conclusion

We can now further estimate that \( x \) falls between \( 0.585 \) and \( 0.588 \). The average of the two may give us a better approximation:

- Midpoint of \( 0.585 \) and \( 0.588 \):
\[
x \approx 0.5865
\]

### Final Estimates

Providing the answer to two decimal places:

The solution to the exponential equation is \( x \approx 0.59 \).