Basic simultaneous equation with logarithms

Question

Question

Solve giving your answers as exact fractions, the simultaneous equations :

$$8^y = 4^{2x + 3} \tag{1}$$

$$\log_2 y = \log_2x + 4 \tag{2}$$

---

I think that the RHS of eq 1 can be split up, I'm hoping that something will fall into place after trying at least.

$$4^{2x + 3} = 4^3(4^{2x})$$

I'm not sure if having $4^{2x}$ on it's own helps here... The problem is that I'm not too sure what I'm trying to achieve with this problem. I get that Its a simultaneous with logarithms. I want to find a term that I can substitute into one of the equations, or some thing that I can subtract.

I can't do $8^y - \log_2 y $ though.

I want to try and get the top equations into $\log_2$ form so that maybe I can start substituting things about, but I can't seem to do that either.

I'm not actually sure how I would express $8^y$ in $\log_2$. It's base 8 isn't it.

Would be cool to get any advice on this, don't really know what to do with it.

Thanks!

Answer 1

For the second equation, the right side can be rewritten as

$$\log_2x+4=\log_2x+\log_216=\log_216x$$

at which point you can eliminate the logarithms. As for the first equation, I'd recommend expressing both sides as a power of $2$.

Answer 2

$$\log_2 y=\log_2x+4$$ $$y=2^4x$$ $$y=16x$$ Plug into first equation

$$8^{16x}=4^{2x+3}$$ $$4^{24x}=4^{2x+3}$$ $$\vdots$$ Can you take it from there?