Why are there two different types of graph for logarithmic functions $\log_a{X}$ for different range of base,i.e., for : $0<a<1$ and $a>1$?

Question

Why are there two different types of graph for logarithmic functions $\log_a X$ for different range of base,i.e; for : $0<a<1$ and $a>1$ ?

Answer 1

From the definition of $\log x$ we can get, $\log_b{x}=\frac{\log{x}}{\log{b}}$. We are plotting $x$ in $x$-axis and $f(x)=\log_b{x}$ in $y$-axis, we will have $f(1)=0$.

Now, when we have $\boxed{\text{Case i:}}~~0<b<1$, for $0<x<1$ we will have $ \frac{\log{x}}{\log{b}}>0$, as $\log$ gives negative value for any input between $0$ and $1$. So, we will have positive value of $f(x)=\log_b{x}$, and for $x>1$ we have $\log{x}>0$ but, $\log{b}<0$, so,$f(x)=\frac{\log{x}}{\log{b}}<0$. Also, $\log{0}\to \infty$, as we go close to $0$, hence, the curve is not touching the $y$ -axis.

Another side, when we have $\boxed{\text{Case ii:}}$ $b>1$, $\log{b}>0$, now as we know $f(1)=\log_b{1}=0$, we need to check for $(i)~0<x<1$ and $(ii)~x>1$, to know the behavior of the graph. In this case, when $0<x<1$, $\log{x}<0$ making $f(x)= \frac{\log{x}}{\log{b}}<0$ and for $x>1$, we have $\log{x}>0$ making $f(x)= \frac{\log{x}}{\log{b}}>0$.

Answer 2

If you multiply a number between zero and one by itself you get a smaller number .... eg half times half is a quarter.....keep multiplying by itself the smaller it gets

If you multiply a number greater than one by itself you get a bigger number .... eg 2 times 2 is 4

So one graph goes down and the other up

Answer 3

Because $\log_a(x) =\dfrac{\log(x)}{\log(a)} $.

If $a > 1$ then $\log(a) > 0$; if $a < 1$ then $\log(a) < 0$.