Certainly! Let's go through each part of the question step-by-step:
1. Calculate [tex]\(\log_5 3125\)[/tex]:
The logarithm [tex]\(\log_b a\)[/tex] is the exponent to which the base [tex]\(b\)[/tex] must be raised to produce the number [tex]\(a\)[/tex]. In this case, we need to find the power to which 5 must be raised to get 3125.
[tex]\( 5^5 = 3125 \)[/tex]
Therefore, [tex]\(\log_5 3125 = 5.0\)[/tex].
2. Calculate [tex]\(\log_{11} 1331\)[/tex]:
Similarly, for [tex]\(\log_b a\)[/tex], the goal is to find the exponent to which the base [tex]\(b\)[/tex] must be raised to get [tex]\(a\)[/tex]. Here, we need to determine the power to which 11 must be raised to get 1331.
[tex]\( 11^3 = 1331 \)[/tex]
So, [tex]\(\log_{11} 1331 = 3.0\)[/tex].
3. Calculate [tex]\(8^3\)[/tex]:
Exponentiation involves raising a base to the power of an exponent. We need to calculate 8 raised to the power of 3.
[tex]\( 8^3 = 8 \times 8 \times 8 = 512 \)[/tex]
Hence, [tex]\(8^3 = 512\)[/tex].
4. Calculate [tex]\(\log_{29} 20736\)[/tex]:
Finally, we need to find [tex]\(\log_{29} 20736\)[/tex], which is the exponent to which the base 29 must be raised to result in 20736.
Here, it can be determined that:
[tex]\( 29^{2.951812701051552} \approx 20736 \)[/tex]
Therefore, [tex]\(\log_{29} 20736 = 2.951812701051552\)[/tex].
With these detailed explanations, we have the results:
[tex]\[
\log_5 3125 = 5.0
\][/tex]
[tex]\[
\log_{11} 1331 = 3.0
\][/tex]
[tex]\[
8^3 = 512
\][/tex]
[tex]\[
\log_{29} 20736 = 2.951812701051552
\][/tex]
