Question

How can logarithms be graphed with different bases? How do transformations affect the logarithmic graph?

Answer 1

Answer:

1) Convert it to the same base and make a table with two columns for x, and y values.

2) The greater the base the lower the curve. Check the picture below.

Step-by-step explanation:

1. Remember the rule in order to convert it to the same base two other logarithms:

$log_{a}b=\frac{log_{c}a}{log_{c} b}$

Example

$log_{5} 20 \\ log_{2} 4\\ \\\frac{log_{10}20}{log_{10}4} =\frac{log2+log10}{log2+log2} = 2,16$

2. When we make the base bigger and bigger the curve will get closer and closer to the y-axis, such as those logarithmic functions.

Algebraically this is why

$y=log_{2}3=1.58$

$y=log_{10}3=0.477$

And so on...

3) Make a table for x values and plug the values to return the y values. Do not forget, x > 0

For $y=log_{2}x$

x   y

1   0

2  1

4  2

$y=log_{10}x$

x  y

1   0

2   0,3

4    0,6

Trace the hyperbole and check for yourself!

Answer 2

To graph with different bases, use the change of base formula.
Transformations may move the parent function.