Answer :
To express \(\log a + \log c + \log f + \log w - \log z\) as a single logarithm, we will use the properties of logarithms, specifically the product and quotient rules.
Step-by-step solution:
1. Combine the sum of logarithms:
We start with the expression \(\log a + \log c + \log f + \log w\).
According to the product rule of logarithms, the sum of logs can be combined into a single log:
[tex]\[ \log a + \log c + \log f + \log w = \log (a \cdot c \cdot f \cdot w) \][/tex]
2. Include the subtraction of a logarithm:
Next, we handle the \(-\log z\) part. According to the quotient rule of logarithms, the subtraction of logs can be written as a division inside a single log:
[tex]\[ \log (a \cdot c \cdot f \cdot w) - \log z = \log \left(\frac{a \cdot c \cdot f \cdot w}{z}\right) \][/tex]
Thus, the expression \(\log a + \log c + \log f + \log w - \log z\) simplifies to a single logarithm:
[tex]\[ \log \left(\frac{a \cdot c \cdot f \cdot w}{z}\right) \][/tex]
So, the final answer is:
[tex]\[ \boxed{\log \left(\frac{a \cdot c \cdot f \cdot w}{z}\right)} \][/tex]
Step-by-step solution:
1. Combine the sum of logarithms:
We start with the expression \(\log a + \log c + \log f + \log w\).
According to the product rule of logarithms, the sum of logs can be combined into a single log:
[tex]\[ \log a + \log c + \log f + \log w = \log (a \cdot c \cdot f \cdot w) \][/tex]
2. Include the subtraction of a logarithm:
Next, we handle the \(-\log z\) part. According to the quotient rule of logarithms, the subtraction of logs can be written as a division inside a single log:
[tex]\[ \log (a \cdot c \cdot f \cdot w) - \log z = \log \left(\frac{a \cdot c \cdot f \cdot w}{z}\right) \][/tex]
Thus, the expression \(\log a + \log c + \log f + \log w - \log z\) simplifies to a single logarithm:
[tex]\[ \log \left(\frac{a \cdot c \cdot f \cdot w}{z}\right) \][/tex]
So, the final answer is:
[tex]\[ \boxed{\log \left(\frac{a \cdot c \cdot f \cdot w}{z}\right)} \][/tex]
