Use the rule logb(xy)=logb x+logb y

Question

to find log2(16⋅64)
.

Answer 1

To find ${\log }_2(16\cdot 64)$ using the rule ${\log }_b(xy)={\log }_{b}x+{\log }_{b}y$, we can break down the expression as follows:

1. Apply the logarithm product rule: $${\log }_2(16\cdot 64)={\log }_2(16)+{\log }_2(64)$$

2. Next, calculate ${\log }_2(16)$: $$16=2^4\Rightarrow {\log }_2(16)=4$$

3. Then calculate ${\log }_2(64)$: $$64=2^6\Rightarrow {\log }_2(64)=6$$

4. Now, combine the results: $${\log }_2(16\cdot 64)={\log }_2(16)+{\log }_2(64)=4+6=10$$

Thus, the value of ${\log }_2(16\cdot 64)$ is $\boxed{{\displaystyle 10}}$.