Answer :
To determine which expression is equivalent to \(\log_2 (9x^3)\), we can use the properties of logarithms.
1. Product Rule of Logarithms: \(\log_b (MN) = \log_b M + \log_b N\)
Applying the product rule to \(\log_2 (9x^3)\):
[tex]\[ \log_2 (9x^3) = \log_2 9 + \log_2 x^3 \][/tex]
2. Power Rule of Logarithms: \(\log_b (M^k) = k \log_b M\)
Applying the power rule to \(\log_2 x^3\):
[tex]\[ \log_2 x^3 = 3 \log_2 x \][/tex]
Now, substitute back into the equation from the product rule:
[tex]\[ \log_2 (9x^3) = \log_2 9 + 3 \log_2 x \][/tex]
Therefore, the expression equivalent to \(\log_2 (9x^3)\) is:
[tex]\[ \log_2 9 + 3 \log_2 x \][/tex]
Hence, the correct answer is:
[tex]\[ \log_2 9 + 3 \log_2 x \][/tex]
1. Product Rule of Logarithms: \(\log_b (MN) = \log_b M + \log_b N\)
Applying the product rule to \(\log_2 (9x^3)\):
[tex]\[ \log_2 (9x^3) = \log_2 9 + \log_2 x^3 \][/tex]
2. Power Rule of Logarithms: \(\log_b (M^k) = k \log_b M\)
Applying the power rule to \(\log_2 x^3\):
[tex]\[ \log_2 x^3 = 3 \log_2 x \][/tex]
Now, substitute back into the equation from the product rule:
[tex]\[ \log_2 (9x^3) = \log_2 9 + 3 \log_2 x \][/tex]
Therefore, the expression equivalent to \(\log_2 (9x^3)\) is:
[tex]\[ \log_2 9 + 3 \log_2 x \][/tex]
Hence, the correct answer is:
[tex]\[ \log_2 9 + 3 \log_2 x \][/tex]
