Last question about logs.

Question

QUESTION

Solve:

$$\log_{\frac 1 2} (x+1)^{ \log_2x+1}=0.3$$

Attempt

$\frac {\log( x+1)}{\log 2}$ multiply $\frac{\log (x+1)}{\log 2^{-1}}$.

Sorry! but thanks.

---

Answer 1

Use this identity: $log(a^{b})=b*log(a)$ for any base.

$$ log_{1/2}((x+1)^{log_{1/2}(x+1)})=log_{1/2}(x+1)*log_{1/2}(x+1)=.3$$

Take the square root of both sides:

$$log_{1/2}(x+1)=\pm(.3)^{1/2}$$

Raise 1/2 to the power of each side:

$$x+1=\left(\frac{1}{2}\right)^{\pm\sqrt{.3}}$$

$$x=-1+2^{\pm\sqrt{.3}}$$