Answer :
Sure, let's evaluate each logarithm step-by-step:
(a) [tex]\(\log_2 2^2\)[/tex]
To find [tex]\(\log_2 2^2\)[/tex], remember that the logarithm asks, "To what power must the base (2) be raised to get 2 squared?" Since [tex]\(2^2\)[/tex] is 4, and 2 raised to the power 2 gives 4, the answer is:
[tex]\[
\log_2 2^2 = 2
\][/tex]
(b) [tex]\(\log_3 81\)[/tex]
We know that 81 can be expressed as a power of 3: [tex]\(81 = 3^4\)[/tex]. Therefore, [tex]\(\log_3 81\)[/tex] asks, "To what power must the base (3) be raised to get 81?" Since [tex]\(3^4\)[/tex] is 81, the answer is:
[tex]\[
\log_3 81 = 4
\][/tex]
(c) [tex]\(\log_5 3125\)[/tex]
Similarly, 3125 can be expressed as a power of 5: [tex]\(3125 = 5^5\)[/tex]. So, [tex]\(\log_5 3125\)[/tex] means, "To what power must the base (5) be raised to get 3125?" Since [tex]\(5^5\)[/tex] is 3125, the answer is:
[tex]\[
\log_5 3125 = 5
\][/tex]
(d) [tex]\(\log_6 6^9\)[/tex]
Here, we have [tex]\(6^9\)[/tex], and the logarithm [tex]\(\log_6 6^9\)[/tex] asks, "To what power must the base (6) be raised to get [tex]\(6^9\)[/tex]?" Since by definition, raising 6 to the power of 9 gives us [tex]\(6^9\)[/tex], the answer is:
[tex]\[
\log_6 6^9 = 9
\][/tex]
So the answers are:
- (a) [tex]\(2\)[/tex]
- (b) [tex]\(4\)[/tex]
- (c) [tex]\(5\)[/tex]
- (d) [tex]\(9\)[/tex]
(a) [tex]\(\log_2 2^2\)[/tex]
To find [tex]\(\log_2 2^2\)[/tex], remember that the logarithm asks, "To what power must the base (2) be raised to get 2 squared?" Since [tex]\(2^2\)[/tex] is 4, and 2 raised to the power 2 gives 4, the answer is:
[tex]\[
\log_2 2^2 = 2
\][/tex]
(b) [tex]\(\log_3 81\)[/tex]
We know that 81 can be expressed as a power of 3: [tex]\(81 = 3^4\)[/tex]. Therefore, [tex]\(\log_3 81\)[/tex] asks, "To what power must the base (3) be raised to get 81?" Since [tex]\(3^4\)[/tex] is 81, the answer is:
[tex]\[
\log_3 81 = 4
\][/tex]
(c) [tex]\(\log_5 3125\)[/tex]
Similarly, 3125 can be expressed as a power of 5: [tex]\(3125 = 5^5\)[/tex]. So, [tex]\(\log_5 3125\)[/tex] means, "To what power must the base (5) be raised to get 3125?" Since [tex]\(5^5\)[/tex] is 3125, the answer is:
[tex]\[
\log_5 3125 = 5
\][/tex]
(d) [tex]\(\log_6 6^9\)[/tex]
Here, we have [tex]\(6^9\)[/tex], and the logarithm [tex]\(\log_6 6^9\)[/tex] asks, "To what power must the base (6) be raised to get [tex]\(6^9\)[/tex]?" Since by definition, raising 6 to the power of 9 gives us [tex]\(6^9\)[/tex], the answer is:
[tex]\[
\log_6 6^9 = 9
\][/tex]
So the answers are:
- (a) [tex]\(2\)[/tex]
- (b) [tex]\(4\)[/tex]
- (c) [tex]\(5\)[/tex]
- (d) [tex]\(9\)[/tex]
