If `log_(3)4=a`,`log_(5)3=b`, then find the value of `log_(3)10` in terms of a and b.

Answer

To find the value of \( \log_{3}10 \) in terms of \( a \) and \( b \), where \( \log_{3}4 = a \) and \( \log_{5}3 = b \), we can follow these steps:

### Step 1: Express \( \log_{3}4 \) in terms of \( \log_{2} \)
We know that:
\[ \log_{3}4 = \log_{3}(2^2) = 2\log_{3}2 \]
Since \( \log_{3}4 = a \), we can write:
\[ 2\log_{3}2 = a \implies \log_{3}2 = \frac{a}{2} \]

**Hint:** Use the property of logarithms that states \( \log_{b}(m^n) = n \cdot \log_{b}(m) \).

### Step 2: Express \( \log_{5}3 \) in terms of \( \log_{3} \)
From the given \( \log_{5}3 = b \), we can express it in terms of base 3:
\[ \log_{5}3 = \frac{\log_{3}3}{\log_{3}5} = \frac{1}{\log_{3}5} \]
Thus, we have:
\[ \log_{3}5 = \frac{1}{b} \]

**Hint:** Use the change of base formula for logarithms: \( \log_{a}b = \frac{1}{\log_{b}a} \).

### Step 3: Express \( \log_{3}10 \)
Now, we can express \( \log_{3}10 \) as:
\[ \log_{3}10 = \log_{3}(2 \cdot 5) = \log_{3}2 + \log_{3}5 \]
Substituting the values we found:
\[ \log_{3}10 = \frac{a}{2} + \frac{1}{b} \]

**Hint:** Use the property of logarithms that states \( \log_{b}(mn) = \log_{b}m + \log_{b}n \).

### Step 4: Combine the terms
To combine the terms, we need a common denominator:
\[ \log_{3}10 = \frac{a}{2} + \frac{1}{b} = \frac{ab + 2}{2b} \]

**Hint:** When adding fractions, find a common denominator to combine them.

### Final Answer
Thus, the value of \( \log_{3}10 \) in terms of \( a \) and \( b \) is:
\[ \log_{3}10 = \frac{ab + 2}{2b} \]