Let's analyze and solve the problem step-by-step. We are given a logarithmic equation and asked to write it in exponential form.
### Step 1: Understand the given logarithmic equation
The given logarithmic equation is:
[tex]\[
\log_2(8\sqrt{2}) = \frac{7}{2}
\][/tex]
### Step 2: Convert the logarithmic equation to exponential form
Recall that if [tex]\(\log_b(x) = y\)[/tex], then this can be converted to exponential form as [tex]\(b^y = x\)[/tex].
Here, the base [tex]\(b\)[/tex] is 2, [tex]\(x\)[/tex] is [tex]\(8\sqrt{2}\)[/tex], and [tex]\(y\)[/tex] is [tex]\(\frac{7}{2}\)[/tex]. Thus, the given logarithmic equation can be written in exponential form as:
[tex]\[
2^{\frac{7}{2}} = 8\sqrt{2}
\][/tex]
### Step 3: Verify the conversion by evaluating each component
We will verify the values for base, result (`8\sqrt{2}`), exponent ([tex]\(\frac{7}{2}\)[/tex]), and the result of the exponential form.
#### 3.1 Verify `8\sqrt{2}`:
[tex]\[
8\sqrt{2} \approx 11.3137
\][/tex]
#### 3.2 Evaluate [tex]\(\frac{7}{2}\)[/tex]:
[tex]\[
\frac{7}{2} = 3.5
\][/tex]
#### 3.3 Calculate [tex]\(2^{3.5}\)[/tex]:
Since [tex]\(3.5\)[/tex] is the exponent, calculate [tex]\(2^{3.5}\)[/tex]:
[tex]\[
2^{3.5} = 2^{7/2} \approx 11.3137
\][/tex]
From the evaluations, we observe:
- [tex]\(8\sqrt{2} \approx 11.3137\)[/tex]
- [tex]\(2^{\frac{7}{2}} \approx 11.3137\)[/tex]
This confirms both sides are equal for the given values.
### Step 4: Conclusion
The logarithmic equation [tex]\(\log_2(8\sqrt{2}) = \frac{7}{2}\)[/tex] in exponential form is:
[tex]\[
2^{\frac{7}{2}} = 8\sqrt{2}
\][/tex]
