Given that; 5log_4 a + 48log_a 4 = a/8 Find "a"?

1 Answer
Jul 22, 2018

a in [255,260]

Explanation:

We will use rules concerning logarithm :
log_b(a)=ln(a)/ln(b)

ln(a) + ln(b)= ln(a*b)

aln(b)=ln(b^a)

Here : 5 log_4(a)+48 log_a(4)=a/8

5ln(a)/ln(4)+ln(4^48)/ln(a)=a/8

5ln(a)^2+ln(4)ln(4^48)=(aln(a)ln(4))/8

Let X=ln(a)<=>a=e^X

5X^2-Xe^Xln(4)/8+48ln(4)^2=0

We can't find any solutions with basic mathematic tools, but because the function f(x)=5ln(x)^2-ln(4)/8xln(x)+48ln(4)^2 is continuous on RR_+^(*), and f(255)~=0.92//f(260)~=-3,68, by theorem, there is a solution a in [255,260].

\0/ Here's our answer !