How do you graph logarithms without a calculator?

Question

I need to graph y = 2log(base3)(x+4)+4

Plot two points and the asymptotes (if any) of the graph of the function.

Answer 1

Take results of powers of the base that you know, such as:

3² = 9

3¹ = 3

Solve for when the contents are this value:

x + 4 = 9

x = 5

That means that the log will give 2 when x=5.

y(x) = 2log,,,(x+4)+4

y(5) = 2·log,,,(5+4)+4

= 2·log,,,(9)+4

= 2·(2)+4

= 8

----

x + 4 = 3

x = -1

y(-1) = 2·log,,,(-1+4)+4

= 2·log,,,(3)+4

= 2·(1)+4

= 6

----

This gives two points:

( 5, 8 )

( -1, 6 )

----

Logarithms have a vertical asymptote on the right side when their contents are zero.

x + 4 = 0

x = -4

So you can draw an asymptote there. You can also find higher values to plot points (like 3³), and you can approximate values between x=-1 and x=-2 by taking roots of 3. They only have approximate values though.