questions on graphs of logarithms and exponentials

Question

1. For this, if i take the log, I know that $\ln (4) > \ln(3)$ The question doesn't specify if $x>0$ but I will assume so. So I have

$-x\ln(4) < -x\ln(3)$ So $g(x)=3^{-x}$ is the "bigger graph." my question is, how do i know which is the "bigger" graph but just looking at the picture?

2. 

I know I have to set the equations equal to each other, so I have:

$3^{2x}-3^x =20$

$(3^x)^2 - 3^x = 20$

$3^x(3^x -1) = 20$

Confused on where to go from here...

Answer 1

Note that $$f(x)=\left(\frac{1}{4}\right)^x\space\text{and}\space g(x)=\left(\frac{1}{3}\right)^x$$

Since successive powers of $\frac{1}{4}$ are smaller than successive powers of $\frac{1}{3},$ $g(x)$ should be the bigger or taller graph for $x>0$, and this is the blue graph. By taller we mean the graph that lies above the other in terms of height.

For the second question, $$3^{2x}-3^x=20\iff (3^x)^2-(3^x)-20=0$$

Now let $u=3^x$ so that $$(3^x)^2-(3^x)-20=0\iff u^2-u-20=0$$

Can you take it from here?

Answer 2

For the first graph, the red one is $y=4^{-x}$ because for positive values of $x$ it is less than $3^{-x}$ and for negative values of $x$ it is larger than $3^{-x}$

For the second graph you have $t^2-t-20=0$ where $t=3^x$

The positive value of $t$ is $5$, that is $3^x=5$ or $x=\log _3 5$