Exponential and logarithmic equations

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Solve the following equations:

(a) $3^{x}=\left(\dfrac{1}{27}\right)^{3}$;

(b) $4^{x}-2^{x+1}-1=0$;

(c) $1-3^{x}+6\left(\dfrac{1}{3}\right)^{x}=0$;

(d) $\log_3(2x-1)=2$;

(e) $\log_2 x=\log_4 x-1$;

(f) $\log_3 x+\log_x 3=\dfrac{10}{3}$.

Solution

(a) $\left(\tfrac1{27}\right)^3=(3^{-3})^3=3^{-9}\Rightarrow x=-9$.

(b) $t=2^x>0$: $t^2-2t-1=0\Rightarrow t=1+\sqrt2$; $x=\log_2(1+\sqrt2)\approx1.272$.

(c) $t=3^x>0$: $1-t+\tfrac6t=0\Rightarrow t^2-t-6=0\Rightarrow t=3\Rightarrow x=1$.

(d) $2x-1=9\Rightarrow x=5$.

(e) $\log_4 x=\tfrac{\log_2 x}{2}$; let $u=\log_2 x$: $u=\tfrac u2-1\Rightarrow u=-2\Rightarrow x=\tfrac14$.

(f) Let $u=\log_3 x$: $u+\tfrac1u=\tfrac{10}3\Rightarrow 3u^2-10u+3=0\Rightarrow u=3$ or $u=\tfrac13\Rightarrow x=27$ or $x=\sqrt[3]3\approx1.442$.

$\boxed{x_a=-9,\ x_b=\log_2(1+\sqrt2)\approx1.27,\ x_c=1,\ x_d=5,\ x_e=\tfrac14,\ x_f\in\{27,\ \sqrt[3]3\}}$