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 exponential term:

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

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

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

Now we can calculate this using approximate logarithmic values. Using a calculator:

- \(\log(6) \approx 0.7781\)
- \(\log(23) \approx 1.3624\)

Substituting these values in gives:

\[
x \approx \frac{0.7781}{1.3624} \approx 0.5715
\]

Now, to provide the solution to two decimal places:

\[
x \approx 0.57
\]

If using an over/under table method, we can find values of \(x\) that make \(23^x\) equal to numbers slightly above and below \(6\) to narrow down the solution:

1. For \(x = 0.5\):
\[
23^{0.5} \approx 4.796
\]
(too low)

2. For \(x = 0.6\):
\[
23^{0.6} \approx 6.659
\]
(too high)

As we see \(x = 0.5\) is too low and \(x = 0.6\) is too high, we can test a value in between:

3. For \(x = 0.57\):
\[
23^{0.57} \approx 6
\]

This shows that our value falls around \(0.57\).

Thus, the final estimated solution to two decimal places is:

\[
\text{The solution to the exponential equation is } x \approx 0.57.
\]