To solve this problem, we need to match each logarithmic expression on the left side with its corresponding value on the right side.
1. [tex]\(\log_{10} x = 3\)[/tex]
- The value of [tex]\(x\)[/tex] for which [tex]\(\log_{10} x = 3\)[/tex] can be found by rewriting the equation in exponential form: [tex]\(x = 10^3\)[/tex].
- Therefore, [tex]\(x = 1000\)[/tex].
- So, [tex]\(\log_{10} x = 3\)[/tex] pairs with [tex]\(1000\)[/tex].
2. [tex]\(\log_{5} x = 4\)[/tex]
- Similarly, rewrite the logarithmic equation as an exponential one: [tex]\(x = 5^4\)[/tex].
- Calculating, [tex]\(5^4 = 625\)[/tex].
- Hence, [tex]\(\log_{5} x = 4\)[/tex] pairs with [tex]\(625\)[/tex].
3. [tex]\(\log_{4} x = 2\)[/tex]
- Again, rewrite the logarithmic equation in its exponential form: [tex]\(x = 4^2\)[/tex].
- Calculating, [tex]\(4^2 = 16\)[/tex].
- There is no 16 in the given set of values, so this logarithmic equation does not pair with any of the provided values.
4. [tex]\(\log_{2} x = 5\)[/tex]
- Rewriting this logarithmic equation in exponential form: [tex]\(x = 2^5\)[/tex].
- Calculating, [tex]\(2^5 = 32\)[/tex].
- Therefore, [tex]\(\log_{2} x = 5\)[/tex] pairs with [tex]\(32\)[/tex].
5. [tex]\(\log_{3} z = 1\)[/tex]
- Rewrite this as an exponential equation: [tex]\(z = 3^1\)[/tex].
- Calculating, [tex]\(3^1 = 3\)[/tex].
- There is no 3 in the given set of values, so this logarithmic equation does not pair with any of the provided values.
Based on these calculations, the correct pairs are:
- [tex]\(\log_{10} x = 3 \longleftrightarrow 1000\)[/tex]
- [tex]\(\log_{5} x = 4 \longleftrightarrow 625\)[/tex]
- [tex]\(\log_{2} x = 5 \longleftrightarrow 32\)[/tex]
Now, filling in the pairs in the box:
[tex]\[
\begin{aligned}
&\log_{10} x = 3 \quad \longleftrightarrow \quad 1000 \\
&\log_{5} x = 4 \quad \longleftrightarrow \quad 625 \\
&\log_{2} x = 5 \quad \longleftrightarrow \quad 32 \\
\end{aligned}
\][/tex]
