To solve the logarithmic equation [tex]\(\log_{\sqrt{3}} x - 8 = 0\)[/tex], follow these steps:
1. Isolate the logarithmic term on one side of the equation:
[tex]\[
\log_{\sqrt{3}} x - 8 = 0
\][/tex]
Add 8 to both sides:
[tex]\[
\log_{\sqrt{3}} x = 8
\][/tex]
2. Rewrite the logarithmic equation in its exponential form:
The logarithmic form [tex]\(\log_{\sqrt{3}} x = 8\)[/tex] can be expressed as:
[tex]\[
x = (\sqrt{3})^8
\][/tex]
3. Calculate [tex]\((\sqrt{3})^8\)[/tex]:
First, note that [tex]\(\sqrt{3}\)[/tex] can be written as [tex]\(3^{1/2}\)[/tex]. So, we have:
[tex]\[
(\sqrt{3})^8 = \left(3^{1/2}\right)^8
\][/tex]
To simplify the exponent, multiply [tex]\(1/2\)[/tex] by [tex]\(8\)[/tex]:
[tex]\[
\left(3^{1/2}\right)^8 = 3^{(1/2) \times 8} = 3^4
\][/tex]
Finally, calculate [tex]\(3^4\)[/tex]:
[tex]\[
3^4 = 81
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = 81
\][/tex]
So, the correct answer is [tex]\(x = 81\)[/tex].
