The value of `log(70/33) + log (22/135) - log(7/18)` is (given log 2 = 0.3010. log 3 = 0.4771)

Options

A. -0.0512
B. 0.4123
C. 0.301
D. None of these

Answer

To solve the expression \( \log\left(\frac{70}{33}\right) + \log\left(\frac{22}{135}\right) - \log\left(\frac{7}{18}\right) \), we can use the properties of logarithms. Here’s a step-by-step solution:

### Step 1: Apply the properties of logarithms
Using the property that \( \log a + \log b = \log(ab) \) and \( \log a - \log b = \log\left(\frac{a}{b}\right) \), we can combine the logarithmic terms.

\[ \log\left(\frac{70}{33}\right) + \log\left(\frac{22}{135}\right) = \log\left(\frac{70 \times 22}{33 \times 135}\right) \]

So, we rewrite the expression as:

\[ \log\left(\frac{70 \times 22}{33 \times 135}\right) - \log\left(\frac{7}{18}\right) \]

Now, we can combine the two logarithmic terms:

\[ \log\left(\frac{70 \times 22}{33 \times 135} \times \frac{18}{7}\right) \]

### Step 2: Simplify the expression inside the logarithm
Now we simplify the expression:

\[ \frac{70 \times 22 \times 18}{33 \times 135 \times 7} \]

### Step 3: Factor and cancel terms
Now let's factor and simplify:

- \( 70 = 2 \times 5 \times 7 \)
- \( 22 = 2 \times 11 \)
- \( 18 = 2 \times 3^2 \)
- \( 33 = 3 \times 11 \)
- \( 135 = 3^3 \times 5 \)

Substituting these values into the expression gives:

\[ \frac{(2 \times 5 \times 7) \times (2 \times 11) \times (2 \times 3^2)}{(3 \times 11) \times (3^3 \times 5) \times 7} \]

Now canceling out the common terms:

- The \( 7 \) in the numerator and denominator cancels out.
- The \( 5 \) in the numerator and denominator cancels out.
- The \( 11 \) in the numerator and denominator cancels out.

This leaves us with:

\[ \frac{2^3 \times 3^2}{3^4} = \frac{8 \times 9}{27} = \frac{72}{27} = \frac{8}{3} \]

### Step 4: Rewrite the logarithm
Now we can write:

\[ \log\left(\frac{8}{3}\right) \]

### Step 5: Use the properties of logarithms again
We can express this as:

\[ \log(8) - \log(3) \]

We know that \( 8 = 2^3 \), so:

\[ \log(8) = \log(2^3) = 3\log(2) \]

Thus, we have:

\[ 3\log(2) - \log(3) \]

### Step 6: Substitute the values of logarithms
Now we substitute the given values \( \log(2) = 0.3010 \) and \( \log(3) = 0.4771 \):

\[ 3(0.3010) - 0.4771 = 0.9030 - 0.4771 = 0.4259 \]

### Final Answer
The value of \( \log\left(\frac{70}{33}\right) + \log\left(\frac{22}{135}\right) - \log\left(\frac{7}{18}\right) \) is:

\[ \boxed{0.4259} \]