To solve the equation [tex]\(\log _{\sqrt{5}}(x) = 2\)[/tex], we will perform the following steps:
1. Understand the meaning of the logarithmic equation:
The equation [tex]\(\log _{\sqrt{5}}(x) = 2\)[/tex] means that [tex]\(\sqrt{5}\)[/tex] raised to the power of [tex]\(2\)[/tex] results in [tex]\(x\)[/tex].
In general, if [tex]\( \log_b(a)=c \)[/tex], it means [tex]\( b^c = a \)[/tex].
2. Convert the logarithmic equation to an exponential form:
Given [tex]\(\log _{\sqrt{5}}(x) = 2\)[/tex], we can rewrite it as:
[tex]\[
(\sqrt{5})^2 = x
\][/tex]
3. Simplify the right-hand side:
We need to calculate [tex]\((\sqrt{5})^2\)[/tex]. By the properties of exponents, we know:
[tex]\[
(\sqrt{5})^2 = 5
\][/tex]
4. Write the final solution:
Hence, we have:
[tex]\[
x = 5
\][/tex]
Therefore, the solution to the equation [tex]\(\log _{\sqrt{5}}(x) = 2\)[/tex] is [tex]\(x = 5\)[/tex].
