Answer :
Let's evaluate each part of the question step-by-step.
(a) [tex]\(\log_6 6^{14}\)[/tex]
When you have a logarithm where the base of the logarithm is the same as the base of the exponent in the argument (inside the logarithm), you can simply take the exponent. This is because of the identity:
[tex]\[
\log_b (b^x) = x
\][/tex]
In this case, since the base is 6 and the exponent is 14, the expression [tex]\(\log_6 6^{14}\)[/tex] simplifies to:
[tex]\[
14
\][/tex]
(b) [tex]\(\log_2 8\)[/tex]
To solve this, we need to express 8 as a power of 2. We know that:
[tex]\[
8 = 2^3
\][/tex]
Now, using the identity [tex]\(\log_b (b^x) = x\)[/tex], we can say:
[tex]\[
\log_2 8 = \log_2 (2^3) = 3
\][/tex]
(c) [tex]\(\log_5 3125\)[/tex]
First, express 3125 as a power of 5. We can rewrite 3125 as:
[tex]\[
3125 = 5^5
\][/tex]
Applying the identity [tex]\(\log_b (b^x) = x\)[/tex], we get:
[tex]\[
\log_5 3125 = \log_5 (5^5) = 5
\][/tex]
(d) [tex]\(\log_9 9^{14}\)[/tex]
Here, just like in part (a), the base of the logarithm is the same as the base of the power in the argument. Using the identity:
[tex]\[
\log_b (b^x) = x
\][/tex]
We find:
[tex]\[
\log_9 9^{14} = 14
\][/tex]
In conclusion, the evaluated results for the expressions are:
(a) 14
(b) 3
(c) 5
(d) 14
(a) [tex]\(\log_6 6^{14}\)[/tex]
When you have a logarithm where the base of the logarithm is the same as the base of the exponent in the argument (inside the logarithm), you can simply take the exponent. This is because of the identity:
[tex]\[
\log_b (b^x) = x
\][/tex]
In this case, since the base is 6 and the exponent is 14, the expression [tex]\(\log_6 6^{14}\)[/tex] simplifies to:
[tex]\[
14
\][/tex]
(b) [tex]\(\log_2 8\)[/tex]
To solve this, we need to express 8 as a power of 2. We know that:
[tex]\[
8 = 2^3
\][/tex]
Now, using the identity [tex]\(\log_b (b^x) = x\)[/tex], we can say:
[tex]\[
\log_2 8 = \log_2 (2^3) = 3
\][/tex]
(c) [tex]\(\log_5 3125\)[/tex]
First, express 3125 as a power of 5. We can rewrite 3125 as:
[tex]\[
3125 = 5^5
\][/tex]
Applying the identity [tex]\(\log_b (b^x) = x\)[/tex], we get:
[tex]\[
\log_5 3125 = \log_5 (5^5) = 5
\][/tex]
(d) [tex]\(\log_9 9^{14}\)[/tex]
Here, just like in part (a), the base of the logarithm is the same as the base of the power in the argument. Using the identity:
[tex]\[
\log_b (b^x) = x
\][/tex]
We find:
[tex]\[
\log_9 9^{14} = 14
\][/tex]
In conclusion, the evaluated results for the expressions are:
(a) 14
(b) 3
(c) 5
(d) 14
