logs :(

\(ln2^{4x-1}=ln8^{x+5}+log_2 16^{1-2x}\)

the right handside \(log_2 16^{1-2x}\) we can make the power (1-2x) next to the log so we could solve for 2^y=16 which will be 4 

\((1-2x)*log_2 16\)

\(log_2 16 = 4\)

4*(1-2x) + \(ln8^{x+5}\)=\(ln2^{4x-1}\)

4-8x+(x+5)*\(ln8\)=(4x-1)*\(ln2\)

\(\frac{4-8x+(x+5)*ln8}{4x-1}=ln2\)

Expressed in terms of ln2 

Further simplify could be done by using calculator 

ln8=2.07944154

4-8x+(x+5)*2.07944154

4-8x+2.079444154x+10.3972077

4-5.920x+10.3972077 

14.3972077-5.920x/4x-1 =ln2 

ln2=0.693147

14.4-5.9x=(4x-1)*0.693147

Rest is just algebra and indeed you will find that 

indeed you will find x=1.73589087 approx: 1.74 

Which works.

I don't think the question wants the x value so just expressed it in ln2 

the answer key says that it should be x=(16ln2+4)/(ln2+8) 

i just dont know how to get there

Find the solution of the equation .... expressing your answer in terms of ln2. 

\(ln2^{4x-1} = ln8^{x+5}+ log_216^{1-2x}\\ \text{I will assume that you mean}\\ ln(2^{4x-1}) = ln(8^{x+5})+ log_2(16^{1-2x})\\ (4x-1)ln(2) =(x+5) ln(8)+ (1-2x)log_2(16)\\ (4x-1)ln(2) =(x+5) ln(2^3)+ (1-2x)log_2(2^4)\\ (4x-1)ln(2) =3(x+5) ln(2)+4 (1-2x)log_2(2)\\ (4x-1)ln(2) =3(x+5) ln(2)+4 (1-2x)\\ [(4x-1)-3(x+5)]ln2=4(1-2x)\\ [4x-1-3x-15)]ln2=4(1-2x)\\ [x-16)]ln2=4(1-2x)\\ xln2-16ln2=4-8x\\ xln2+8x=4+16ln2\\ x(ln2+8)=\\ x=\frac{4(1+4ln2)}{ln2+8}\\ x=\frac{16ln2+4}{ln2+8}\) 

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Coding:

ln2^{4x-1} = ln8^{x+5}+ log_216^{1-2x}\\
\text{I will assume that you mean}\\
ln(2^{4x-1}) = ln(8^{x+5})+ log_2(16^{1-2x})\\
(4x-1)ln(2) =(x+5) ln(8)+ (1-2x)log_2(16)\\
(4x-1)ln(2) =(x+5) ln(2^3)+ (1-2x)log_2(2^4)\\
(4x-1)ln(2) =3(x+5) ln(2)+4 (1-2x)log_2(2)\\
(4x-1)ln(2) =3(x+5) ln(2)+4 (1-2x)\\
[(4x-1)-3(x+5)]ln2=4(1-2x)\\
[4x-1-3x-15)]ln2=4(1-2x)\\
[x-16)]ln2=4(1-2x)\\
xln2-16ln2=4-8x\\
xln2+8x=4+16ln2\\
x(ln2+8)=\\
x=\frac{4(1+4ln2)}{ln2+8}\\
x=\frac{16ln2+4}{ln2+8}