Question

Use an example to show that $\log (a+b)\ne \log \left(a\right)+\log \left(b\right)$

Short answer

By using the example of $\log \left(12\right)\ne \log \left(5\right)+\log \left(7\right)$, it has been proved that $\log (a+b)\ne \log \left(a\right)+\log \left(b\right)$

Step-by-step solution

Step 1. To give an example for the property

To give an example for the property, $\log (a+b)\ne \log \left(a\right)+\log \left(b\right)$

Step 2. Substitute two numbers.

Consider $$\begin{gathered} a=5, \\ b=7 \end{gathered}$$for the logarithmic equation.

$\log (5+7)=\log \left(5\right)+\log \left(7\right)$

Step 3. Solve the equation

$$\begin{gathered} \log (5+7)=\log \left(5\right)+\log \left(7\right) \\ \log \left(12\right)=\log \left(5\right)+\log \left(7\right) \end{gathered}$$

Step 4. Substitute the logarithmic values and solve.

$$\begin{gathered} \log \left(12\right)=\log \left(5\right)+\log \left(7\right) \\ 1.079=0.699+0.845 \\ 1.079\ne 1.544 \end{gathered}$$

Hence the example.