Certainly! Let's solve the given expression step-by-step: [tex]\(2 \log_7 \left(\sqrt[7]{343}\right)\)[/tex].
1. Identify the inner expression:
[tex]\[
\sqrt[7]{343}
\][/tex]
This is the 7th root of 343.
2. Write 343 as a power of 7:
[tex]\[
343 = 7^3
\][/tex]
So we rewrite the inner expression as:
[tex]\[
\sqrt[7]{7^3} = (7^3)^{1/7}
\][/tex]
3. Simplify the exponent:
[tex]\[
(7^3)^{1/7} = 7^{3/7}
\][/tex]
4. Take the logarithm:
The logarithm expression becomes:
[tex]\[
\log_7 \left(7^{3/7}\right)
\][/tex]
Using the logarithmic property [tex]\(\log_b (a^c) = c \cdot \log_b (a)\)[/tex], where [tex]\(b\)[/tex] is the base, [tex]\(a\)[/tex] is the argument, and [tex]\(c\)[/tex] is the exponent, we have:
[tex]\[
\log_7 \left(7^{3/7}\right) = \frac{3}{7} \log_7 (7)
\][/tex]
5. Simplify using the identity [tex]\(\log_b (b) = 1\)[/tex]:
Since [tex]\(\log_7 (7) = 1\)[/tex], the expression becomes:
[tex]\[
\frac{3}{7} \cdot 1 = \frac{3}{7}
\][/tex]
6. Multiply by 2:
Finally, we need to multiply this result by 2:
[tex]\[
2 \cdot \frac{3}{7} = \frac{6}{7}
\][/tex]
Converting the fraction to a decimal for the final result, [tex]\(\frac{6}{7} \approx 0.8571428571428571\)[/tex].
Therefore, the value of the expression [tex]\(2 \log_7 \left(\sqrt[7]{343}\right)\)[/tex] is approximately [tex]\(0.8571428571428571\)[/tex].
