To solve the problem of simplifying the expression [tex]\(\log_3 27 + \log_3 3\)[/tex] into a single logarithm, we can use the properties of logarithms, specifically the product rule. The product rule of logarithms states that:
[tex]\[
\log_b (MN) = \log_b M + \log_b N
\][/tex]
where [tex]\(b\)[/tex] is the base of the logarithm, and [tex]\(M\)[/tex] and [tex]\(N\)[/tex] are the numbers involved.
Given the expression:
[tex]\[
\log_3 27 + \log_3 3
\][/tex]
we can apply the product rule of logarithms. According to the product rule, we can combine the two logarithms into a single logarithm as follows:
[tex]\[
\log_3 27 + \log_3 3 = \log_3 (27 \cdot 3)
\][/tex]
Next, we simplify [tex]\(27 \cdot 3\)[/tex]:
[tex]\[
27 \cdot 3 = 81
\][/tex]
So, the expression becomes:
[tex]\[
\log_3 (27 \cdot 3) = \log_3 81
\][/tex]
We now have a single logarithm:
[tex]\[
\log_3 81
\][/tex]
To fully simplify, we recognize that [tex]\(81\)[/tex] is a power of [tex]\(3\)[/tex]. Specifically:
[tex]\[
81 = 3^4
\][/tex]
Thus,
[tex]\[
\log_3 81 = \log_3 (3^4)
\][/tex]
By using the power rule of logarithms, which states [tex]\(\log_b (a^c) = c \cdot \log_b a\)[/tex], we get:
[tex]\[
\log_3 (3^4) = 4 \cdot \log_3 3
\][/tex]
Since [tex]\(\log_3 3\)[/tex] is 1 (because any logarithm of a number to its own base is 1), we have:
[tex]\[
4 \cdot \log_3 3 = 4 \cdot 1 = 4
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\log_3 27 + \log_3 3 = 4
\][/tex]
So,
[tex]\[
\log_3 27 + \log_3 3 = 4
\][/tex]
