Let `log_(10)2=a` and `log_(10)3=b` determine the following in term of `a` and `b`
` log_(4)100+2log_(27)100`

Answer

To solve the problem, we need to express \( \log_4 100 + 2\log_{27} 100 \) in terms of \( a \) and \( b \), where \( a = \log_{10} 2 \) and \( b = \log_{10} 3 \).

### Step-by-Step Solution:

1. **Convert the logarithms to base 10**:
We can use the change of base formula for logarithms:
\[ \log_b x = \frac{\log_k x}{\log_k b} \]
Therefore,
\[ \log_4 100 = \frac{\log_{10} 100}{\log_{10} 4} \]
and
\[ \log_{27} 100 = \frac{\log_{10} 100}{\log_{10} 27} \]

2. **Simplify \( \log_{10} 100 \)**:
Since \( 100 = 10^2 \),
\[ \log_{10} 100 = 2 \]

3. **Substituting back**:
Now substituting \( \log_{10} 100 \) into the expressions:
\[ \log_4 100 = \frac{2}{\log_{10} 4} \]
and
\[ \log_{27} 100 = \frac{2}{\log_{10} 27} \]

4. **Express \( \log_{10} 4 \) and \( \log_{10} 27 \)**:
We can express \( \log_{10} 4 \) and \( \log_{10} 27 \) in terms of \( a \) and \( b \):
\[ \log_{10} 4 = \log_{10} (2^2) = 2 \log_{10} 2 = 2a \]
and
\[ \log_{10} 27 = \log_{10} (3^3) = 3 \log_{10} 3 = 3b \]

5. **Substituting these into the logarithm expressions**:
Now substituting these values back into our expressions:
\[ \log_4 100 = \frac{2}{2a} = \frac{1}{a} \]
and
\[ \log_{27} 100 = \frac{2}{3b} \]

6. **Combine the terms**:
Now we can substitute these into the original expression:
\[ \log_4 100 + 2\log_{27} 100 = \frac{1}{a} + 2 \cdot \frac{2}{3b} = \frac{1}{a} + \frac{4}{3b} \]

### Final Result:
Thus, the expression \( \log_4 100 + 2\log_{27} 100 \) in terms of \( a \) and \( b \) is:
\[ \frac{1}{a} + \frac{4}{3b} \]