The expression into a single logarithm is [tex]log[(x)^{10}][(2)^{30}][/tex]
Step-by-step explanation:
Let us revise some logarithmic rules
- [tex]log(a)^{n}=nlog(a)[/tex]
- [tex]log(ab)=log(a)+log(b)[/tex]
- [tex]nlog(a)+mlog(b)=log[(a)^{n}][(b)^{m}][/tex]
∵ 10 log(x) + 5 log(64)
- At first re-write 10 log(x)
∴ 10 log(x) = [tex]log(x)^{10}[/tex]
- Then re-write 5 log(64)
∴ 5 log(64) = [tex]log(64)^{5}[/tex]
∴ 10 log(x) + 5 log(64) = [tex]log(x)^{10}[/tex] + [tex]log(64)^{5}[/tex]
- Use the 3rd rule above to make it single logarithm
∵ [tex]log(x)^{10}[/tex] + [tex]log(64)^{5}[/tex] = [tex]log[(x)^{10}][(64)^{5}][/tex]
∴ 10 log(x) + 5 log(64) = [tex]log[(x)^{10}][(64)^{5}][/tex]
∵ 64 = 2 × 2 × 2 × 2 × 2 × 2
∴ We can write 64 as [tex]2^{6}[/tex]
∴ [tex](64)^{5}=(2^{6})^{5}[/tex]
- Multiply the two powers of 2
∴ [tex](64)^{5}=(2)^{30}[/tex]
∴ 10 log(x) + 5 log(64) = [tex]log[(x)^{10}][(2)^{30}][/tex]
The expression into a single logarithm is [tex]log[(x)^{10}][(2)^{30}][/tex]
Learn more:
You can learn more about the logarithmic functions in brainly.com/question/11921476
#LearnwithBrainly
