For the exponential function e^x and logarithmic function log x, graphically show the effect if x is halved. Include a table of values for all four functions.
Could someone please explain to me what occurs when things are halved.
e^x becomes e^(x/2) is x is halved
ln x becomes ln(x/2) is x is halved.
Start out with x = 4 and compare what you get with x = 2 and x = 1.
ln4 = 1.386
ln2 = 0.693
ln1 = 0
Each time x doubles, ln x increases by 0.693
e^4 = 54.60
e^2 = 7.389
e^1 = e = 2.718
Each time x doubles, e^x gets squared.
Can you all be a tutor for me? I'm a quick learner!
When an input value is halved, it means that it is divided by 2. To analyze the effect of halving on the exponential function e^x and the logarithmic function log(x), we can compare the graphs and values for different inputs.
Exponential Function (e^x):
To graphically show the effect of halving on the exponential function e^x, we can create a table of values and observe the changes. Let's assume we start with the input values x = 0, x = 1, x = 2, and so on.
Table for e^x:
```
x | e^x
------------
0 | 1
1 | e
2 | e^2
... | ...
```
Now, let's halve the input values by dividing them by 2:
Table for e^x (with halved x):
```
x/2 | e^(x/2)
--------------
0 | e^0 = 1
0.5 | e^(0.5)
1 | e^1 = e
1.5 | e^(1.5)
... | ...
```
As x/2 gets larger (0.5, 1, 1.5, etc.), the corresponding values of e^(x/2) will also increase. However, the rate of increase will be slower compared to when x is not halved.
Logarithmic Function (log x):
For the logarithmic function log(x), we will again create a table of values and observe the changes when x is halved.
Table for log x:
```
x | log x
------------
1 | 0
10 | 1
100 | 2
... | ...
```
Now, let's halve the input values:
Table for log x (with halved x):
```
x/2 | log (x/2)
-----------------
0.5 | log 0.5
5 | log 5
50 | log 50
... | ...
```
As x/2 gets smaller (0.5, 5, 50, etc.), the corresponding values of log(x/2) will decrease. Again, the rate of decrease will be slower compared to when x is not halved.
In summary, for the exponential function e^x, halving the input x/2 leads to a slower rate of increase in the output values. For the logarithmic function log x, halving the input x/2 leads to a slower rate of decrease in the output values.