## Simplification of Logarithms

## Example

### Question

Simplify (without a calculator): \(3\log3+\log125\)

### Apply the appropriate logarithmic laws to simplify the expression

\begin{align*} 3\log3+\log125 &= 3\log3+\log{5^{3}} \\ &= 3\log3 + 3\log{5} \\ &= 3 ( \log3 + \log{5} ) \\ &= 3 \log{(3 \times 5)} \\ &= 3 \log{15} \end{align*}

### Write the final answer

We cannot simplify any further, therefore \(3\log3+\log125 = 3 \log{15}\).

**Important:** All the algebraic manipulation techniques \((\times, \div, +, -\), factorisation etc.) also apply for logarithmic expressions. Always be aware of the number of terms in an expression as this will help to determine how to simplify.