Answer :
Sure! Let's evaluate each of these logarithmic expressions step-by-step:
(a) [tex]\(\log_2 2^{11}\)[/tex]
To evaluate this expression, we can use the logarithmic identity:
[tex]\[
\log_b (b^x) = x
\][/tex]
Here, the base [tex]\(b\)[/tex] is 2, and the exponent [tex]\(x\)[/tex] is 11. So, [tex]\(\log_2 2^{11} = 11\)[/tex].
(b) [tex]\(\log_3 243\)[/tex]
First, we need to express 243 as a power of 3. Since [tex]\(3^5 = 243\)[/tex], we apply the property of logarithms:
[tex]\[
\log_b (b^x) = x
\][/tex]
Substituting the values, [tex]\(\log_3 243 = 5\)[/tex].
(c) [tex]\(\log_5 3125\)[/tex]
Similarly, we need to express 3125 as a power of 5. Since [tex]\(5^5 = 3125\)[/tex], we can use the same logarithmic property:
[tex]\[
\log_b (b^x) = x
\][/tex]
So, [tex]\(\log_5 3125 = 5\)[/tex].
(d) [tex]\(\log_6 6^7\)[/tex]
Using the property:
[tex]\[
\log_b (b^x) = x
\][/tex]
Here, the base [tex]\(b\)[/tex] is 6, and the exponent [tex]\(x\)[/tex] is 7. Thus, [tex]\(\log_6 6^7 = 7\)[/tex].
So, the evaluated expressions are:
- (a) [tex]\(\log_2 2^{11} = 11\)[/tex]
- (b) [tex]\(\log_3 243 = 5\)[/tex]
- (c) [tex]\(\log_5 3125 = 5\)[/tex]
- (d) [tex]\(\log_6 6^7 = 7\)[/tex]
(a) [tex]\(\log_2 2^{11}\)[/tex]
To evaluate this expression, we can use the logarithmic identity:
[tex]\[
\log_b (b^x) = x
\][/tex]
Here, the base [tex]\(b\)[/tex] is 2, and the exponent [tex]\(x\)[/tex] is 11. So, [tex]\(\log_2 2^{11} = 11\)[/tex].
(b) [tex]\(\log_3 243\)[/tex]
First, we need to express 243 as a power of 3. Since [tex]\(3^5 = 243\)[/tex], we apply the property of logarithms:
[tex]\[
\log_b (b^x) = x
\][/tex]
Substituting the values, [tex]\(\log_3 243 = 5\)[/tex].
(c) [tex]\(\log_5 3125\)[/tex]
Similarly, we need to express 3125 as a power of 5. Since [tex]\(5^5 = 3125\)[/tex], we can use the same logarithmic property:
[tex]\[
\log_b (b^x) = x
\][/tex]
So, [tex]\(\log_5 3125 = 5\)[/tex].
(d) [tex]\(\log_6 6^7\)[/tex]
Using the property:
[tex]\[
\log_b (b^x) = x
\][/tex]
Here, the base [tex]\(b\)[/tex] is 6, and the exponent [tex]\(x\)[/tex] is 7. Thus, [tex]\(\log_6 6^7 = 7\)[/tex].
So, the evaluated expressions are:
- (a) [tex]\(\log_2 2^{11} = 11\)[/tex]
- (b) [tex]\(\log_3 243 = 5\)[/tex]
- (c) [tex]\(\log_5 3125 = 5\)[/tex]
- (d) [tex]\(\log_6 6^7 = 7\)[/tex]
