To determine the transformed logarithmic equation that passes through the given points [tex]\((1,0)\)[/tex] and [tex]\((0,3)\)[/tex] using the equation [tex]\( y = a \log ( \pm x + c) \)[/tex], we will follow these steps.
### Step 1: Set Up the General Equation
The general form of the equation is:
[tex]\[ y = a \log (\pm x + c) \][/tex]
### Step 2: Substitute the First Point [tex]\((1,0)\)[/tex] into the Equation
For the point [tex]\((1,0)\)[/tex], substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = a \log (\pm 1 + c) \][/tex]
This simplifies to:
[tex]\[ 0 = a \log (\pm 1 + c) \][/tex]
Since [tex]\( a \neq 0 \)[/tex], this implies:
[tex]\[ \log (\pm 1 + c) = 0 \][/tex]
[tex]\[ \pm 1 + c = 1 \][/tex]
Therefore, [tex]\( c = 1 - 1 = 0 \)[/tex] or [tex]\( c = 1 + 1 = 2 \)[/tex].
So we get two possibilities:
[tex]\[ \pm 1 + c = 1 \Rightarrow c = 0 \quad \text{or} \quad c = 2 \][/tex]
### Step 3: Substitute the Second Point [tex]\((0,3)\)[/tex] into the Equation
For the point [tex]\((0,3)\)[/tex], substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 = a \log (\pm 0 + c) \][/tex]
This simplifies to:
[tex]\[ 3 = a \log (c) \][/tex]
### Step 4: Solve for Constants [tex]\( a \)[/tex] and [tex]\( c \)[/tex]
#### Case 1: [tex]\( c = 0 \)[/tex]
If [tex]\( c = 0 \)[/tex], the logarithmic term [tex]\( \log(c) \)[/tex] is undefined since [tex]\(\log(0)\)[/tex] is not defined. So, [tex]\( c = 0 \)[/tex] is not a valid solution.
#### Case 2: [tex]\( c = 2 \)[/tex]
If [tex]\( c = 2 \)[/tex], then we have:
[tex]\[ 3 = a \log (2) \][/tex]
[tex]\[ a = \frac{3}{\log 2} \][/tex]
### Step 5: Write the Final Equation
So, substituting [tex]\( a \)[/tex] and [tex]\( c \)[/tex] back into the general form, we get:
[tex]\[ y = \frac{3}{\log 2} \log (x + 2) \][/tex]
Therefore, the equation of the transformed logarithm that passes through the points [tex]\((1, 0)\)[/tex] and [tex]\((0, 3)\)[/tex] is:
[tex]\[ f(x) = \frac{3}{\log 2} \log (x + 2) \][/tex]
