Question

Solve. 90^x = 27. Explain every step.

Answer 1

`x \approx0.732`

EXPLANATION

The given exponential equation is

`{90}^(x) = 27`

We take natural log of both sides

`ln( {90}^(x) ) = ln(27)`

Recall and the following property of logarithms.

`ln( {a}^(n) ) = n \: ln(a)`

This implies that;

`x ln(90) = ln(27)`

Solve for x.

`x = ( ln(27) )/( ln(90) )`

`x \approx0.732`

Answer 2

The answer is ⇒ x = 0.73244

Explanation:

∵ 90^x = 27 ⇒ insert log in both sides

∴
`log(90)^(x)=log(27)`

∵ 90 = 9 × 10 , 27 = 3³

∵ log(a)^b = b log(a)

∴ log(9 × 10)^x = x log(9 × 10)

∴
`xlog(9*10)=log(3^(3))`

∵ log(a × b) = log(a) + log(b)⇒log(9 × 10) = log(9) + log(10)

∴
`x[log(9)+log(10)]=3log(3)`

∵ log(10) = 1 , 9 = 3²

∴
`x[log(3)^(2)+1]=3log(3)`

∴
`x[2log(3)+1] = 3log(3)`

∴ x = (3log3)/[2log(3)+1]

∴ x = 0.73244