Answer :
To express [tex]3\log_5(2) + \log_5(7)[/tex] as a single logarithm, we need to use the properties of logarithms.
Step 1: Use the power rule of logarithms.
The power rule states that [tex]a \cdot \log_b(x) = \log_b(x^a)[/tex].
Apply the power rule to [tex]3\log_5(2)[/tex]:
[tex]3\log_5(2) = \log_5(2^3)[/tex]
This simplifies to:
[tex]\log_5(2^3) = \log_5(8)[/tex]
Step 2: Use the product rule of logarithms.
The product rule states that [tex]\log_b(x) + \log_b(y) = \log_b(xy)[/tex].
Now, combine [tex]\log_5(8)[/tex] and [tex]\log_5(7)[/tex] using the product rule:
[tex]\log_5(8) + \log_5(7) = \log_5(8 \cdot 7)[/tex]
Step 3: Simplify the expression inside the logarithm:
[tex]8 \cdot 7 = 56[/tex]
So, we have:
[tex]\log_5(56)[/tex]
Thus, [tex]3\log_5(2) + \log_5(7)[/tex] expressed as a single logarithm is [tex]\log_5(56)[/tex].
Therefore, the answer is option a) [tex]\log_5(56)[/tex].