Question

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

Answer 1

Answer:

The answer is ⇒ x = 0.73244

Step-by-step 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

Answer 2

ANSWER

$x \approx0.732$

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.

$x = \frac{ ln(27) }{ ln(90) }$

$x \approx0.732$