Compare the graphs of the logarithmic functions f(x)=log7x and g(x)=log4x. For what values of x is f=g, fg, and f

Question

Compare the graphs of the logarithmic functions f(x)=log7x and g(x)=log4x.  For what values of x is f=g, f>g, and f<g?  Explain how you know.

solomonzelman · Answer

Do you agree with this?

$f(x)=\log(7x)=\log(7)+\log(x)$
$g(x)=\log(4x)=\log(4)+\log(x)$

solomonzelman · Answer

This is because $\log(a\times b)=\log(a)+\log(b)$.        Right?

solomonzelman · Answer

So, if you want to compare them (assuming that $\log(x)$ is defined,
that is $x>0$), then you have:

${\tiny~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}f(x)~~■~~g(x)$ $\[0.8em]$
           ${\tiny~~}\log(7x)~~■~~\log(4x)$ $\[0.8em]$
$\log(7)+\log(x)~~■~~\log(4)+\log(x)$ $\[0.8em]$
             ${\tiny~~~}\log(7)~~■~~\log(4)$ $\[0.8em]$
(And note that this comparison is for any value of $x$)

jadareese_18 · Answer

I think I'm following but a little confused with the a and b part.

solomonzelman · Answer

I am using $ ■$ just as an unknown symbol $-,~+,~\le,~\ge,~=$ etc.

(Similarly and for the same purpose, the symbol $ ♣$ is in use.)

jadareese_18 · Answer

Also the numbers after the (log) part of the equation were suppose to be lowered.  I wasn't sure how to do that.

solomonzelman · Answer

Oh, that is just a general rule, that for any positive A and B, if you have a log of a product of A and B, then this is equivalent to the sum of individual logs of A and of B.

or, mathematically, $\log(a\times b)=\log(a)+\log(b)$.

solomonzelman · Answer

What do you mean that "the (log) part of the equation were suppose to be lowered" ?

jadareese_18 · Answer

the number after the log part so it should have been f(x)=log v7 x and g(x)=log v4 x where the 7 and 4 are lowered below the log and x.

solomonzelman · Answer

Oh ... I see.

solomonzelman · Answer

So they are not logs of 7x and of 4x.
Rather, they are log base 7 of x, and base 4 of x.

solomonzelman · Answer

$f(x)=\log_7x \ g(x)=\log_4x$

jadareese_18 · Answer

Yes that is correct.  I'm sorry.

solomonzelman · Answer

If you are familiar with latex/tex, then here is the code to use.
`$f(x)=\log_7x$`   (You make a subscript with $\color{red}{\text{_}}$ or with $\color{red}{\text{_{ }}}$.

jadareese_18 · Answer

Oh thank you!  That's really nice to know.

solomonzelman · Answer

$f(x)=\log_7x \quad \Longleftrightarrow \quad7^{f(x)}=x\quad \Longleftrightarrow \quad f^{-1}(x)=7^{x}$.
$g(x)=\log_4x \quad \Longleftrightarrow \quad4^{g(x)}=x\quad \Longleftrightarrow \quad g^{-1}(x)=4^{x}$.

We can compare their inverse functions, that might be easier.|dw:1478559623286:dw|