What is the value of `log_(10)(9/8)-log_(10)((27)/(32))+log_(10)(3/4)` ?

Options

A. 3
B. 2
C. 1
D. 0

Answer

To solve the expression \( \log_{10}\left(\frac{9}{8}\right) - \log_{10}\left(\frac{27}{32}\right) + \log_{10}\left(\frac{3}{4}\right) \), we can use the properties of logarithms. Here are the steps:

### Step 1: Combine the logarithms
Using the property of logarithms that states \( \log_a(b) + \log_a(c) = \log_a(b \cdot c) \), we can combine the first two logarithms:

\[ \log_{10}\left(\frac{9}{8}\right) + \log_{10}\left(\frac{3}{4}\right) = \log_{10}\left(\frac{9}{8} \cdot \frac{3}{4}\right) \]

### Step 2: Calculate the product inside the logarithm
Now, calculate the product:

\[ \frac{9}{8} \cdot \frac{3}{4} = \frac{9 \cdot 3}{8 \cdot 4} = \frac{27}{32} \]

### Step 3: Substitute back into the logarithm
Now we can substitute this back into the logarithm:

\[ \log_{10}\left(\frac{27}{32}\right) - \log_{10}\left(\frac{27}{32}\right) \]

### Step 4: Use the property of logarithms for subtraction
Using the property that states \( \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right) \):

\[ \log_{10}\left(\frac{27}{32}\right) - \log_{10}\left(\frac{27}{32}\right) = \log_{10}\left(1\right) \]

### Step 5: Evaluate the logarithm
Since \( \log_{10}(1) = 0 \):

\[ \log_{10}(1) = 0 \]

### Final Answer
Thus, the value of the expression \( \log_{10}\left(\frac{9}{8}\right) - \log_{10}\left(\frac{27}{32}\right) + \log_{10}\left(\frac{3}{4}\right) \) is:

\[ \boxed{0} \]

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