Question

Find the value of yy in each equation. Explain how you determined the value of y.

a. y = 3^log3(3)
b. y = 3^log3(9)
c. y = 3^log3(81)
d. y = 3^log3(x)

Answer 1

a) 3

b) 9

c) 81

d) x

Explanation:

We know the properties of log function as:

1) log(AB) = log(A) + log(B)

2)
`\log((A)/(B)) = \log(A)+\log(B)`

3) log(aᵇ) = b × log(a)

also,

4)
`\log_b(a)=(\log(a))/(\log(b))`

Given:

a. y =
`3^(\log_3(3))`

Now,

taking log both sides, we get

log(y) =
`\log(3^(\log_3(3)))`

using 3, we get

log(y) = log₃(3) × log(3)

using 4, we get

log(y) =
`(\log(3))/(\log(3))` × log(3)

or

log(y) = 1 × log(3)

taking anti-log both sides

y = 3

b. y =
`3^(log_3(9))`

Now,

taking log both sides, we get

log(y) =
`\log(3^(\log_3(9)))`

using 3, we get

log(y) = log₃(9) × log(3)

using 4, we get

log(y) =
`(\log(9))/(\log(3))` × log(3)

or

log(y) = log(9)

taking anti-log both sides

y = 9

c. y =
`3^(\log_3(81))`

Now,

taking log both sides, we get

log(y) =
`\log(3^(\log_3(81)))`

using 3, we get

log(y) = log₃(81) × log(3)

using 4, we get

log(y) =
`(\log(81))/(\log(3))` × log(3)

or

log(y) = log(81)

taking anti-log both sides

y = 81

d. y =
`3^(\log_3(x))`

Now,

taking log both sides, we get

log(y) =
`\log(3^(\log_3(x)))`

using 3, we get

log(y) = log₃(x) × log(3)

using 4, we get

log(y) =
`(\log(x))/(\log(3))` × log(3)

or

log(y) = log(x)

taking anti-log both sides

y = x