exponents
Its graph looks like
We can evalute this function:
When we evaluate a function, we know the domain number and we seek its range partner. In this case, we know and we seek the pair , where is the value of the function at .
Reverse
We can also think in reverse.
Here, we know the value of the function. We seek the domain numbers paired with it. We know , we seek the pair , where is the solution to the equation.
For , as long as we give a positive function value, then we can find the associated
domain number.
For , every positive function value has exactly one associated domain number. That
is eeriely backwards of the function rule.
Remember: Each domain number in a function is paired with exactly one range number. In other words, a function can have only one value at each domain number.
We just said a similar sentence for the range numbers of an exponential function.
We just said that each range number for an exponential function is paired with exactly one domain number.
That’s not a requirement to be a function. It is an extra characteristics of exponential functions.
Thinking of an exponential function in reverse sounds like a new function. It is. We call it a logarithmic function.
The logarithm function just reverses the pairs in the exponential function. If is a pair in the exponential function, then is a pair in the logarithmic function.
Therefore, the domain and range switch.
Logarithm
It seems weird to write , even though it is perfectly correct.
We are used to representing the domain number by a letter, , and then the function value as a formula involving .
We would prefer our pairs in the logarithm function to look like
What’s the expression?
Logarithmic functions come from the study of logarithms, which gets shortened to
log for notation purposes.
In our example here, we are working with the base exponential, . So, the reverse is called the logarithm base . We tack on a subscript to complete the name of our new function.
That pairs in the logarithmic base function look like
The coordinates of the points on the graph look like
is just the exponent that needs to equal .
Remember: are the reverse of the exponential pairs : .
is also for some .
is the number that you raise to, to get .
We have a logarithmic function for every exponential function. They are designated by their bases.
Since all of the pairs are reversed, the graphs switch axes.
The horizontal axis is an asymptote for the basic exponential function. Now, the vertical axis is an asymptote for the basic logarithmic function.
The graph of a basic exponential function has as an intercept. The graph of a basic logarithmic function has as an intercept.
Their graphs are mirror images of each other across the diagonal through quadrants I and III.
From this basic logarithmic function and its graph we can analyze and transform for more general logarithmic functions.
Analyze
The inside of the logarithm is and this equals when . This has to be the vertical asymptote.
The inside of the logarithm is , and this is positive for . The graph must move to the right.
The leading coefficient is , which is positive. Therefore, the graph hugs the asymptote up down the asymptote.
when . Therefore, the anchor point has moved to . . This gives us the point .
Graph of .
Our analysis tells us that:
- The implied domain of is .
- The implied range of is .
- is always increasing decreasing .
- has no maximums or minimums.
We can also see that the graph will have a horizontal intercept, which means the
function has a zero.
Remember: is the thing that you raise to, to get and . Therefore, is the thing that you raise to, to get . must be .
is a logarithmic function, so it must have a partner exponential function. The pairs for look like or just . The pairs for the exponential function would look like . The roles of and would be switched. would be the variable in the formula. would be the function value.
We can obtain the formula for this partner exponential function by solving the logarithmic formula for .
Remember: is the thing that you raise to, to get and . Therefore, is the thing that you raise to, to get
Here is the graph of both the logarithmic and the associated exponential functions.
Analyze
The inside of the logarithm is and this equals when . has to be the vertical asymptote.
The inside of the logarithm is , and this is positive for . The graph must move to the right.
The leading coefficient is , which is positive negative . Therefore, the graph is flipped vertically from the basic graph. It goes up the asymptote.
when . Therefore, the anchor point has moved to .
Graph of .
- The natural or implied domain of is .
- The implied range of is .
- is always increasing.
- has no maximums or minimums.
Reversing all of the pairs will give the associated exponential function.
is a logarithmic function, so it must have a partner exponential function. The pairs for look like . The pairs for the exponential function would look like . The roles of and would be switched. would be the variable in the exponential formula. would be the exponential function value.
We can obtain the formula for this partner exponential function by solving the logarithmic formula for .
Remember: is the thing that you raise to, to get and . Therefore, is the thing that you raise to, to get
Here is the graphs of both the logarithmic and the associated exponential functions.
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more examples can be found by following this link
More Examples of Percent Change