(a) Find x given that 4^x=2 (b) Let x = 1.1609... be the positive real number such that 4^x=5. Prove that x is irrational.

Question

(a) Find x given that
`4^x=2`

(b) Let x = 1.1609... be the positive real number such that
`4^x=5`. Prove that x is irrational.

Answer 1

Answers:

(a) x = 0.5

(b) Given Below.

Step-by-step explanation:

(a)

`4^x = 2`

apply ln both sides

`ln(4^x) = ln(2)`

simplify

`xln(4) = ln(2)`

divide both sides by ln(4)

`x = 0.5`

(b)

Recall that rational numbers can be expressed in fractions a/b

`4^x = 5`

apply ln on both sides

`ln( 4^x) = ln(5)`

simplify

`xln( 4) = ln(5)`

divide both sides by ln(4)

`x =( ln(5))/(ln(4))`

simplify

`x =1.160964047...` The value is endless and continues.

The following cannot be expressed as a fraction or whole number.

Hence, the following value of x is irrational.