To determine which of the given functions is a logarithmic function, let's analyze the characteristics of each option:
1. Option 1: [tex]\( y = \log_3 x \)[/tex]
This is a logarithmic function because it expresses [tex]\( y \)[/tex] as the logarithm of [tex]\( x \)[/tex] to the base 3. The general form of a logarithmic function is [tex]\( y = \log_b x \)[/tex], where [tex]\( b \)[/tex] is the base of the logarithm. Here, [tex]\( b = 3 \)[/tex].
2. Option 2: [tex]\( y = 3^x \)[/tex]
This is an exponential function, not a logarithmic function. The general form of an exponential function is [tex]\( y = b^x \)[/tex], where [tex]\( b \)[/tex] is the base. Here, [tex]\( b = 3 \)[/tex].
3. Option 3: [tex]\( y = x + 3 \)[/tex]
This is a linear function. It presents [tex]\( y \)[/tex] as a linear combination of [tex]\( x \)[/tex]. The general form for a linear function is [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept. Here, the slope [tex]\( m = 1 \)[/tex] and the y-intercept [tex]\( c = 3 \)[/tex].
4. Option 4: [tex]\( y = x^3 \)[/tex]
This is a power function, specifically a cubic function. It expresses [tex]\( y \)[/tex] as a polynomial of degree 3. The general form for a polynomial function is [tex]\( y = x^n \)[/tex], where [tex]\( n \)[/tex] is the degree of the polynomial. Here, [tex]\( n = 3 \)[/tex].
Based on the analysis, the function that fits the form of a logarithmic function is:
Option 1: [tex]\( y = \log_3 x \)[/tex]
Hence, [tex]\( y = \log_3 x \)[/tex] is the logarithmic function.
