Question

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

Answer 1

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) $\log(\frac{A}{B}) = \log(A)+\log(B)$

3) log(aᵇ) = b × log(a)

also,

4) $\log_b(a)=\frac{\log(a)}{\log(b)}$

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) =  $\frac{\log(3)}{\log(3)}$ × 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) =  $\frac{\log(9)}{\log(3)}$ × 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) =  $\frac{\log(81)}{\log(3)}$ × 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) =  $\frac{\log(x)}{\log(3)}$ × log(3)

or

log(y) =  log(x)

taking anti-log both sides

y = x