To match each logarithmic equation to its corresponding [tex]\( x \)[/tex]-value, let's analyze each equation step-by-step.
1. We have the equation [tex]\(\log_5 x = 4\)[/tex]. To find [tex]\( x \)[/tex], we solve for [tex]\( x \)[/tex] using the properties of logarithms:
[tex]\[
x = 5^4 = 625
\][/tex]
So, [tex]\(\log_5 x = 4 \rightarrow 625\)[/tex].
2. Next, we have the equation [tex]\(\log_{10} x = 3\)[/tex]. Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = 10^3 = 1000
\][/tex]
So, [tex]\(\log_{10} x = 3 \rightarrow 1000\)[/tex].
3. For the equation [tex]\(\log_4 x = 2\)[/tex], we solve for [tex]\( x \)[/tex]:
[tex]\[
x = 4^2 = 16
\][/tex]
So, [tex]\(\log_4 x = 2 \rightarrow 16\)[/tex].
4. Now we have the equation [tex]\(\log_3 x = 1\)[/tex]. Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = 3^1 = 3
\][/tex]
So, [tex]\(\log_3 x = 1 \rightarrow 3\)[/tex].
5. Finally, we have the equation [tex]\(\log_2 x = 5\)[/tex]. Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 2^5 = 32
\][/tex]
So, [tex]\(\log_2 x = 5 \rightarrow 32\)[/tex].
Now we can form the correct pairs:
[tex]\[
\begin{aligned}
& \log_5 x = 4 \longleftrightarrow 625, \\
& \log_{10} x = 3 \longleftrightarrow 1000, \\
& \log_4 x = 2 \longleftrightarrow 16, \\
& \log_3 x = 1 \longleftrightarrow 3, \\
& \log_2 x = 5 \longleftrightarrow 32. \\
\end{aligned}
\][/tex]
So the correct pairs are:
[tex]\[
\log_5 x = 4 \longleftrightarrow 625 \\
\log_{10} x = 3 \longleftrightarrow 1000 \\
\log_4 x = 2 \longleftrightarrow 16 \\
\log_3 x = 1 \longleftrightarrow 3 \\
\log_2 x = 5 \longleftrightarrow 32
\][/tex]
These pairs correctly match each logarithmic equation to its corresponding [tex]\( x \)[/tex]-value.
