Question

The vertical asymptote of the function f(x) = 3 log(x + 3) is x = what

Answer 1

you shifted the function \(\large\color{slate}{ \displaystyle y=\log(x) }\) which way ?

(disregard the coefficient in front for this problem)

Answer 2

idk I added 3? @SolomonZelman

Answer 3

yes, \(\large\color{slate}{ \displaystyle f(x+c) }\) is a shift to the left by c units from \(\large\color{slate}{ \displaystyle f(x) }\).

Answer 4

so x is 3? @SolomonZelman

Answer 5

where would the asymptote for \(\normalsize \color{blue}{ \displaystyle y=\log(x) }\) normally be?

Answer 6

take some number that approaches zero from the right side
(such that it is getting 0.1 to 0.023 to 0.0007 to 0.00001 to 0.0000...0001 )
doesn't have to use those values particularly, but the point is that it is getting closer and closer to zero.

Answer 7

would it make sense to think of such a thing when the number is closer and close approaching zero, as of \(\large\color{black}{ \displaystyle \frac{1}{\infty } }\) ?

Answer 8

and infinity symbol is there to represent a very large number.
because 0.0000...0001 is essentially `1/(really big number)`

Answer 9

am I making sense

Answer 10

?

Answer 11

(so far)

Answer 12

yeah most of it does @SolomonZelman

Answer 13

yes,
now you know the rule for logarithms that
\(\large\color{black}{ \displaystyle \log(a^b)=b\cdot \log(a) }\)

and you know that \(\large\color{black}{ \displaystyle \frac{1}{n}=n^{-1} }\)

Answer 14

So, as you are taking the log of very small number, watch what happens...

Answer 15

\(\large\color{black}{ \displaystyle \log({\rm very~~small~~number}) }\)

\(\large\color{black}{ \displaystyle \log(\frac{1}{{\rm very~~big~~number}}) }\)

\(\large\color{black}{ \displaystyle \log(~~({\rm very~~big~~number})^{-1}~~) }\)

\(\large\color{black}{ \displaystyle (-1)\cdot \log({\rm very~~big~~number}) }\)

\(\large\color{blue}{ \displaystyle -\log({\rm very~~big~~number}) }\)

Answer 16

as this "small number" getting smaller and smaller (saying as it closer and closer approaches zero), the big number (at this time) which is on the bottom is getting larger and larger.

so in the result, as this "small number" is closer to zero, the "very big number" is going more and more towards infinity.

AS WE APPLY THE RULES, we have -log( very big number)
meaning, we get something along the lines of \(\large\color{black}{ \displaystyle -\log(\infty) }\) WHICH would be just as good as saying \(\large\color{black}{ \displaystyle -\infty }\)

Answer 17

now, we can watch on the graph, https://www.desmos.com/calculator/txcoxfbguq

Answer 18

you can see it never hits zero, but the close x to zero - the y is going more towards \(\large\color{black}{ \displaystyle -\infty }\)

Answer 19

when you shift this function y=log(x) 3 units left, you are shifting it's asymptote of x=0
and your new asymptote is?