Logarithmic equations (IV)

Statement

Solve the following equations after stating the existence conditions:

• (a) $\log_3(x)=2$
• (b) $2\log_4(x+2)=2$
• (c) $\log_4(x)-\log_2(x)=1$

Solution

(a) EC $x>0$. $x=3^2=9$.

(b) EC $x>-2$. $\log_4(x+2)=1 \Rightarrow x+2=4 \Rightarrow x=2$.

(c) EC $x>0$. Since $\log_4 x=\tfrac12\log_2 x$: $\tfrac12\log_2 x-\log_2 x=-\tfrac12\log_2 x=1 \Rightarrow \log_2 x=-2 \Rightarrow x=2^{-2}=\tfrac14$.

$\boxed{x_a=9;\quad x_b=2;\quad x_c=\tfrac14}$