Question #3624b

2 Answers
Jun 13, 2017

x=log2532500

Explanation:

52x325x+2=0
52x3=25x+2
Taking the natural logarithm of both sides,
ln(52x3)=ln(25x+2)
(2x3)ln5=(5x+2)ln2
(2ln5)x3ln5=(5ln2)x+2ln2
(2ln5)x(5ln2)x=3ln5+2ln2
[ln(52)ln(25)]x=ln(53)+ln(22)
x=ln(5322)ln(5225)
x=ln(1254)ln(2532)
x=log2532500

Jun 13, 2017

Given: 52x325x+2=0

Move the second term to the right:

52x3=25x+2

Use the base 5 logarithm on both sides:

log5(52x3)=log5(25x+2)

Use the identity logb(ac)=(c)logb(a) on both sides:

(2x3)log5(5)=(5x+2)log5(2)

Use the property logb(b)=1 on the left:

2x3=(5x+2)log5(2)

Use the distributive property on the right:

2x3=5log5(2)x+2log5(2)

Subtract 5log5(2)x from both sides:

(25log5(2))x3=2log5(2)

Add 3 to both sides:

(25log5(2))x=3+2log5(2)

Divide both sides by the coefficient of x:

x=3+2log5(2)25log5(2)

Convert to base e by using the conversion formula log5(x)=ln(x)ln(5):

x=3+2ln(2)ln(5)25ln(2)ln(5)

Multiply by 1 in the form of ln(5)ln(5):

x=3ln(5)+2ln(2)2ln(5)5ln(2)