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≈
0.77
.
Remaining Attempts : 1
to two decimal places.(1 point)
The solution to the exponential equation is x≈
0.77
.
Remaining Attempts : 1
Answers
GPT-4o mini
To estimate the solution to the exponential equation \( 3 + 2^{3x} = 9 \), we can rearrange it into a form suitable for making an over/under table. First, let’s isolate \( 2^{3x} \):
\[
2^{3x} = 9 - 3
\]
\[
2^{3x} = 6
\]
Next, we take the log base 2 of both sides (or use natural logarithm and apply some change of base if needed):
\[
3x = \log_2(6)
\]
Now we can solve for \( x \):
\[
x = \frac{\log_2(6)}{3}
\]
To make it practical, we can estimate \( \log_2(6) \) using values we know. We can calculate \( \log_2(6) \) by recognizing it can be rewritten in terms of common logarithms:
\[
\log_2(6) = \frac{\log_{10}(6)}{\log_{10}(2)}
\]
Using approximate values:
- \( \log_{10}(6) \approx 0.7781 \)
- \( \log_{10}(2) \approx 0.3010 \)
Calculating \( \log_2(6) \):
\[
\log_2(6) \approx \frac{0.7781}{0.3010} \approx 2.585
\]
Now dividing by 3 to find \( x \):
\[
x \approx \frac{2.585}{3} \approx 0.8617
\]
### Creating an Over/Under Table
To estimate and refine our solution further, we can create an over/under table.
We test values of \( x \):
| \( x \) | \( 3 + 2^{3x} \) | Over/Under |
|----------|------------------|------------|
| 0.70 | \( 3 + 2^{3(0.70)} = 3 + 2^{2.1} \approx 3 + 4.25 = 7.25 \) | Over |
| 0.75 | \( 3 + 2^{3(0.75)} = 3 + 2^{2.25} \approx 3 + 4.76 = 7.76 \) | Over |
| 0.80 | \( 3 + 2^{3(0.80)} = 3 + 2^{2.4} \approx 3 + 5.32 = 8.32 \) | Over |
| 0.76 | \( 3 + 2^{3(0.76)} = 3 + 2^{2.28} \approx 3 + 4.97 = 7.97 \) | Over |
| 0.73 | \( 3 + 2^{3(0.73)} = 3 + 2^{2.19} \approx 3 + 4.48 = 7.48 \) | Under |
| 0.74 | \( 3 + 2^{3(0.74)} = 3 + 2^{2.22} \approx 3 + 4.64 = 7.64 \) | Over |
From this analysis, it looks like \( x \) is somewhere between \( 0.73 \) and \( 0.76 \), but closer to 0.75 based on our evaluations.
Thus, we can narrow it down a little further and finalize an estimate:
**Final estimate:**
The solution to the exponential equation \( 3 + 2^{3x} = 9 \) is approximately \( x \approx 0.77 \).
\[
2^{3x} = 9 - 3
\]
\[
2^{3x} = 6
\]
Next, we take the log base 2 of both sides (or use natural logarithm and apply some change of base if needed):
\[
3x = \log_2(6)
\]
Now we can solve for \( x \):
\[
x = \frac{\log_2(6)}{3}
\]
To make it practical, we can estimate \( \log_2(6) \) using values we know. We can calculate \( \log_2(6) \) by recognizing it can be rewritten in terms of common logarithms:
\[
\log_2(6) = \frac{\log_{10}(6)}{\log_{10}(2)}
\]
Using approximate values:
- \( \log_{10}(6) \approx 0.7781 \)
- \( \log_{10}(2) \approx 0.3010 \)
Calculating \( \log_2(6) \):
\[
\log_2(6) \approx \frac{0.7781}{0.3010} \approx 2.585
\]
Now dividing by 3 to find \( x \):
\[
x \approx \frac{2.585}{3} \approx 0.8617
\]
### Creating an Over/Under Table
To estimate and refine our solution further, we can create an over/under table.
We test values of \( x \):
| \( x \) | \( 3 + 2^{3x} \) | Over/Under |
|----------|------------------|------------|
| 0.70 | \( 3 + 2^{3(0.70)} = 3 + 2^{2.1} \approx 3 + 4.25 = 7.25 \) | Over |
| 0.75 | \( 3 + 2^{3(0.75)} = 3 + 2^{2.25} \approx 3 + 4.76 = 7.76 \) | Over |
| 0.80 | \( 3 + 2^{3(0.80)} = 3 + 2^{2.4} \approx 3 + 5.32 = 8.32 \) | Over |
| 0.76 | \( 3 + 2^{3(0.76)} = 3 + 2^{2.28} \approx 3 + 4.97 = 7.97 \) | Over |
| 0.73 | \( 3 + 2^{3(0.73)} = 3 + 2^{2.19} \approx 3 + 4.48 = 7.48 \) | Under |
| 0.74 | \( 3 + 2^{3(0.74)} = 3 + 2^{2.22} \approx 3 + 4.64 = 7.64 \) | Over |
From this analysis, it looks like \( x \) is somewhere between \( 0.73 \) and \( 0.76 \), but closer to 0.75 based on our evaluations.
Thus, we can narrow it down a little further and finalize an estimate:
**Final estimate:**
The solution to the exponential equation \( 3 + 2^{3x} = 9 \) is approximately \( x \approx 0.77 \).