Answer:
a) 3
b) 9
c) 81
d) x
Step-by-step explanation:
We know the properties of log function as:
1) log(AB) = log(A) + log(B)
2) [tex]\log(\frac{A}{B}) = \log(A)+\log(B)[/tex]
3) log(aᵇ) = b × log(a)
also,
4) [tex]\log_b(a)=\frac{\log(a)}{\log(b)}[/tex]
Given:
a. y = [tex]3^{\log_3(3)}[/tex]
Now,
taking log both sides, we get
log(y) = [tex]\log(3^{\log_3(3)})[/tex]
using 3, we get
log(y) = log₃(3) × log(3)
using 4, we get
log(y) = [tex]\frac{\log(3)}{\log(3)}[/tex] × log(3)
or
log(y) = 1 × log(3)
taking anti-log both sides
y = 3
b. y = [tex]3^{log_3(9)}[/tex]
Now,
taking log both sides, we get
log(y) = [tex]\log(3^{\log_3(9)})[/tex]
using 3, we get
log(y) = log₃(9) × log(3)
using 4, we get
log(y) = [tex]\frac{\log(9)}{\log(3)}[/tex] × log(3)
or
log(y) = log(9)
taking anti-log both sides
y = 9
c. y = [tex]3^{\log_3(81)}[/tex]
Now,
taking log both sides, we get
log(y) = [tex]\log(3^{\log_3(81)})[/tex]
using 3, we get
log(y) = log₃(81) × log(3)
using 4, we get
log(y) = [tex]\frac{\log(81)}{\log(3)}[/tex] × log(3)
or
log(y) = log(81)
taking anti-log both sides
y = 81
d. y = [tex]3^{\log_3(x)}[/tex]
Now,
taking log both sides, we get
log(y) = [tex]\log(3^{\log_3(x)})[/tex]
using 3, we get
log(y) = log₃(x) × log(3)
using 4, we get
log(y) = [tex]\frac{\log(x)}{\log(3)}[/tex] × log(3)
or
log(y) = log(x)
taking anti-log both sides
y = x
