Certainly! Let's solve the expression [tex]\(\log_2 8 - \log_3 \sqrt{27} + \log_7 49\)[/tex] step-by-step.
1. Calculate [tex]\(\log_2 8\)[/tex]:
The expression [tex]\(\log_2 8\)[/tex] asks for the power to which 2 must be raised to obtain 8. Since [tex]\(2^3 = 8\)[/tex], we have:
[tex]\[
\log_2 8 = 3
\][/tex]
2. Calculate [tex]\(\log_3 \sqrt{27}\)[/tex]:
First, simplify [tex]\(\sqrt{27}\)[/tex]. The square root of 27 can be written as:
[tex]\[
\sqrt{27} = 27^{1/2}
\][/tex]
Since [tex]\(27 = 3^3\)[/tex], we can rewrite [tex]\(\sqrt{27}\)[/tex] as:
[tex]\[
\sqrt{27} = (3^3)^{1/2} = 3^{3/2}
\][/tex]
Thus, [tex]\(\log_3 \sqrt{27}\)[/tex] becomes:
[tex]\[
\log_3 (3^{3/2}) = \frac{3}{2}
\][/tex]
3. Calculate [tex]\(\log_7 49\)[/tex]:
The expression [tex]\(\log_7 49\)[/tex] asks for the power to which 7 must be raised to obtain 49. Since [tex]\(7^2 = 49\)[/tex], we have:
[tex]\[
\log_7 49 = 2
\][/tex]
4. Combine the results:
Now, substitute the calculated values back into the original expression:
[tex]\[
\log_2 8 - \log_3 \sqrt{27} + \log_7 49 = 3 - \frac{3}{2} + 2
\][/tex]
Simplify the expression step-by-step:
[tex]\[
3 - \frac{3}{2} = 3 - 1.5 = 1.5
\][/tex]
Then add:
[tex]\[
1.5 + 2 = 3.5
\][/tex]
So, the value of the original expression [tex]\(\log_2 8 - \log_3 \sqrt{27} + \log_7 49\)[/tex] is:
[tex]\[
\boxed{3.5}
\][/tex]
