Use the rule logb(xy)=logbx+logby

Question

log
𝑏
⁡
(
𝑥
𝑦
)
=
log
𝑏
⁡
𝑥
+
log
𝑏
⁡
𝑦
to find log2(16⋅64)
log
2
⁡
(
16
⋅
64
)

Answer 1

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

1. First, calculate ${\log }_2(16)$ and ${\log }_2(64)$.

   • Since $16=2^4$, we have: $${\log }_2(16)={\log }_2(2^4)=4$$

   • Since $64=2^6$, we have: $${\log }_2(64)={\log }_2(2^6)=6$$

2. Now use the property of logarithms: $${\log }_2(16\cdot 64)={\log }_2(16)+{\log }_2(64)$$ Substituting the values we found: $${\log }_2(16\cdot 64)=4+6=10$$

Therefore, ${\log }_2(16\cdot 64)=10$.