Question

1. the expression $3 + \log_{2} x$ can be written as the single logarithm:

a. $3\log_{2} x$
b. $\log_{2} x^{3}$
c. $\log_{2}(8x)$
d. $\log_{2}(9x)$

Explanation:

Step1: Recall log property \( n = \log_b b^n \)

We know that for any base \( b \), \( n=\log_b(b^n) \). Here, \( b = 2 \) and \( n = 3 \), so \( 3=\log_2(2^3)=\log_2(8) \).

Step2: Use log addition rule \( \log_b a+\log_b c=\log_b(ac) \)

The original expression is \( 3 + \log_2 x \), substitute \( 3=\log_2 8 \) into it, we get \( \log_2 8+\log_2 x \).
By the logarithm addition rule \( \log_b a+\log_b c = \log_b(ac) \), where \( a = 8 \), \( b = 2 \), \( c = x \), so \( \log_2 8+\log_2 x=\log_2(8\times x)=\log_2(8x) \).

Answer:

C. \( \log_2(8x) \)