logarithmic equations for beginners

Question

I just started learning logarithms and I'm having problems with one equations.

$\log_{\sqrt{x}}{2} + \log_{x}{4} = 4$

Sorry if I'm writing this wrong just for understanding ,I'm using this form log(b,a).

Answer 1

Hint

Use that

$$\log_{\sqrt{x}} 2=\frac{\log 2}{\log \sqrt x}=\frac{\log 2}{\log x^{1/2}}=\frac{\log 2}{1/2\log x}=\frac{2\log 2}{\log x}=\frac{\log 2^2}{\log x}=\frac{\log 4}{\log x}$$ and

$$\log_x 4=\frac{\log 4}{\log x}.$$

Answer 2

You need to convert everything to the same base.

1.) log(sqrt(x),2) can be rewritten as 0.5*log(x,2) 2.) log(x,4) is the same as log(x,2)/log(x,4) which is again 0.5*log(x,2)

The equation then reduces to log(x,2) = 4 which gives x as 16