Please explain the steps for this problem: 2 ln x + 3 ln 2 = 5?

Answer 1

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

First, use the log properties to change the natural logs. Use this log property:

log[base b](a^c) = c*log[base b](a).

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

ln(x^2) + ln(8) = 5

Now, use this log property:
log[base b](a) + log[base b](c) = log[base b](ac)

ln(8x^2) = 5

Now, change this to exponential form. This is done by realizing
log[base b](a) = c if and only if b^c = a

e^5 = 8x^2

Now, solve for x.

(1/8)e^5 = x^2, so

x = +/- sqrt( e^5 / 8)

We can reduce that to

x = +/- [(e^2)/2] sqrt(e / 2)

So our potential answer is

x = { [(e^2)/2] sqrt(e / 2) , - [(e^2)/2] sqrt(e / 2) }

But we reject the negative solution because if we attempt to plug it into our original equation, we will end up taking the log of a negative number (which is not allowed). Therefore we only have one solution, and

x = [(e^2)/2] sqrt(e / 2)

Answer 2

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

By one of the laws of logs, a ln(b) = ln(a^b)
So the equation becomes :
ln(x^2) + ln(2^3) = 5
or,
ln(x^2) + ln(8) = 5

By another law of logs, ln(a) + ln(b) = ln(a*b)
So the equation becomes :
ln(8x^2) = 5

Now, ln(e) = 1, so, multiplying through by 5 gives :
5ln(e) = 5. Again using the first log rule, ln(e^5) = 5.

So now we have :
ln(8x^2) = ln(e^5)

Taking the antilog of both sides gives:
8x^2 = e^5
so, x^2 = e^5 / 8

Therefore, x = sqrt(e^5 / 8) = 4.30716204...

Answer 3

you use two formulas ln a + ln b = ln ab and ln a^b = b lna


2lnx = lnx^2 and 3 ln 2 = ln2^3=ln8

so 2 ln x + 3 ln 2 = ln 8x^2 =5

you use exponentials 8x^2 = e^5

and x = (e^5/8) ^0.5 =4.3

Answer 4

ln x² + ln 2³ = 5

ln(2³. x²) = 5

2³.x² = e^(5)

x ² = (1/8) . e^(5)

x = ± (1/√8).[ e^(5/2) ]

Answer 5

2 ln x + 3 ln 2 = 5
2ln x =5 - 3 ln 2
ln x = 5/2 -3/2 ln 2
e^lnx = e^(5/2 -3/2 ln 2)
x = (e^5/2)(e^-3/2ln2)=(e^5/2)/e^ln2^3/2 = (e^5/2)/2^3/2
=(e^5/2)/(2sqrt(2))

Answer 6

Take the inverse natural log (e) of both sides and solve for x. It's easier than you think.

Answer 7

x=e^(5/2) * (rad2)/4