Exponential equations and a graph

Text

Solve the following:

• (a) solve $4^x - 6\cdot 2^x + 8 = 0$;
• (b) solve $3^{x+1} = 7$ (leave the answer as a logarithm);
• (c) sketch $y = -2\cdot \left(\dfrac{1}{3}\right)^x + 4$, giving the asymptote, two points (a table) and whether it increases or decreases.

Solution

(a) With $t=2^x>0$: $t^2-6t+8=0\Rightarrow t=2\ \vee\ t=4$, so $x=1$ and $x=2$.

(b) $x+1=\log_3 7\Rightarrow x=\log_3 7-1\approx0{,}771$.

(c) Asymptote $y=4$; $\left(\tfrac13\right)^x$ decreases but the $-2$ coefficient flips it → increasing, below the asymptote. Points $(0;2)$, $(1;\tfrac{10}{3})$. $x$-intercept: $\left(\tfrac13\right)^x=2\Rightarrow x=\log_{1/3}2\approx-0{,}63$.

$\boxed{\text{(a) } x=1,2;\ \text{(b) } x=\log_3 7-1;\ \text{(c) } x_0\approx-0{,}63,\ y=4}$