What is x if 3ln2+ln(x^2)+2=43ln2+ln(x2)+2=4?

2 Answers
Nov 8, 2015

x=e^{1-3/2 ln(2)}x=e132ln(2)

Explanation:

Isolate the term involving xx:

ln(x^2) = 4-2-3ln(2) = 2-3ln(2)ln(x2)=423ln(2)=23ln(2)

Use the property of logarithm ln(a^b)=bln(a)ln(ab)=bln(a):

2ln(x)=2-3ln(2)2ln(x)=23ln(2)

Isolate the term involving xx again:

ln(x)=1-3/2 ln(2)ln(x)=132ln(2)

Take the exponential of both terms:

e^{ln(x)} = e^{1-3/2 ln(2)}eln(x)=e132ln(2)

Consider the fact that exponential and logarithm are inverse functions, and thus e^{ln(x)} =xeln(x)=x

x=e^{1-3/2 ln(2)}x=e132ln(2)

Nov 8, 2015

x=+-(esqrt2)/4x=±e24

Explanation:

[1]" "3ln2+ln(x^2)+2=4[1] 3ln2+ln(x2)+2=4

Subtract 22 from both sides.

[2]" "3ln2+ln(x^2)+2-2=4-2[2] 3ln2+ln(x2)+22=42

[3]" "3ln2+ln(x^2)=2[3] 3ln2+ln(x2)=2

Property: alog_bm=log_bm^aalogbm=logbma

[4]" "ln2^3+ln(x^2)=2[4] ln23+ln(x2)=2

[5]" "ln8+ln(x^2)=2[5] ln8+ln(x2)=2

Property: log_bm+log_bn=log_b(mn)logbm+logbn=logb(mn)

[6]" "ln(8x^2)=2[6] ln(8x2)=2

[7]" "log_e(8x^2)=2[7] loge(8x2)=2

Convert to exponential form.

[8]" "hArre^2=8x^2[8] e2=8x2

Divide both sides by 88.

[9]" "e^2/8=x^2[9] e28=x2

Subtract e^2/8e28 from both sides.

[10]" "x^2-e^2/8=0[10] x2e28=0

Difference of two squares.

[11]" "(x+sqrt(e^2/8))(x-sqrt(e^2/8))=0[11] (x+e28)(xe28)=0

[12]" "(x+e/(2sqrt2))(x-e/(2sqrt2))=0[12] (x+e22)(xe22)=0

Rationalize.

[13]" "(x+(esqrt2)/4)(x-(esqrt2)/4)=0[13] (x+e24)(xe24)=0

Therefore: color(blue)(x=+-(esqrt2)/4)x=±e24