To evaluate the expression [tex]\(\log_3(9^{400})\)[/tex] using the Laws of Logarithms, follow these steps:
1. Understand the base of the exponent in the argument: Recognize that [tex]\(9\)[/tex] can be expressed as a power of [tex]\(3\)[/tex]. Specifically, [tex]\(9\)[/tex] is the same as [tex]\(3^2\)[/tex]. Therefore, the expression inside the logarithm can be rewritten:
[tex]\[
9^{400} = (3^2)^{400}
\][/tex]
2. Use the power rule for exponents: According to the power rule, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Applying this gives:
[tex]\[
(3^2)^{400} = 3^{2 \cdot 400} = 3^{800}
\][/tex]
3. Apply the logarithm to the simplified expression: Now, the expression becomes:
[tex]\[
\log_3(3^{800})
\][/tex]
4. Use the power rule of logarithms: The power rule for logarithms states that [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex]. Here, let [tex]\(a = 3\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(c = 800\)[/tex]. Therefore:
[tex]\[
\log_3(3^{800}) = 800 \cdot \log_3(3)
\][/tex]
5. Simplify the logarithm: Recall that [tex]\(\log_b(b) = 1\)[/tex] for any base [tex]\(b\)[/tex] because any number raised to the power of 1 is itself. Thus:
[tex]\[
\log_3(3) = 1
\][/tex]
6. Finalize the calculation: Substitute [tex]\(\log_3(3) = 1\)[/tex] back into the expression:
[tex]\[
800 \cdot \log_3(3) = 800 \cdot 1
\][/tex]
Thus, the final result is:
[tex]\[
800
\][/tex]
Hence, the evaluated value of the expression [tex]\(\log_3(9^{400})\)[/tex] is [tex]\(800\)[/tex].
