Answer :
To simplify the expression [tex]\( \log 64 - \log 128 + \log 32 \)[/tex], let's apply the properties of logarithms step by step.
### Step 1: Use the property [tex]\(\log{a} - \log{b} = \log{\left(\frac{a}{b}\right)}\)[/tex]
This property allows us to combine the logarithms with subtraction into a single logarithm. Applying it to the first part of the expression:
[tex]\[ \log 64 - \log 128 = \log{\left(\frac{64}{128}\right)} \][/tex]
Simplify the fraction inside the logarithm:
[tex]\[ \frac{64}{128} = \frac{1}{2} \][/tex]
Therefore:
[tex]\[ \log 64 - \log 128 = \log{\left(\frac{1}{2}\right)} \][/tex]
### Step 2: Use the property [tex]\(\log{a} + \log{b} = \log{(a \cdot b)}\)[/tex]
Now we need to combine this result with [tex]\(\log 32\)[/tex]:
[tex]\[ \log{\left(\frac{1}{2}\right)} + \log 32 \][/tex]
Using the property for addition of logarithms, we get:
[tex]\[ \log{\left(\frac{1}{2} \times 32\right)} = \log{(16)} \][/tex]
### Result:
After simplifying the expression using logarithmic properties, we find that:
[tex]\[ \log 64 - \log 128 + \log 32 = \log 16 \][/tex]
This is the simplified form of the given logarithmic expression.
### Step 1: Use the property [tex]\(\log{a} - \log{b} = \log{\left(\frac{a}{b}\right)}\)[/tex]
This property allows us to combine the logarithms with subtraction into a single logarithm. Applying it to the first part of the expression:
[tex]\[ \log 64 - \log 128 = \log{\left(\frac{64}{128}\right)} \][/tex]
Simplify the fraction inside the logarithm:
[tex]\[ \frac{64}{128} = \frac{1}{2} \][/tex]
Therefore:
[tex]\[ \log 64 - \log 128 = \log{\left(\frac{1}{2}\right)} \][/tex]
### Step 2: Use the property [tex]\(\log{a} + \log{b} = \log{(a \cdot b)}\)[/tex]
Now we need to combine this result with [tex]\(\log 32\)[/tex]:
[tex]\[ \log{\left(\frac{1}{2}\right)} + \log 32 \][/tex]
Using the property for addition of logarithms, we get:
[tex]\[ \log{\left(\frac{1}{2} \times 32\right)} = \log{(16)} \][/tex]
### Result:
After simplifying the expression using logarithmic properties, we find that:
[tex]\[ \log 64 - \log 128 + \log 32 = \log 16 \][/tex]
This is the simplified form of the given logarithmic expression.
