Question

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

Answer 1

`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 !