# Relationship between logarithm and exponents definition

### Logarithm - Wikipedia

b) What is the relationship between the graph of y = e x and the graph Rule i) embodies the definition of a logarithm: log bx is the exponent to which b must be . In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which More explicitly, the defining relation between exponentiation and logarithm is: log b ( x) = y. understand the relationship between the exponential function f(x) = ex and the This example demonstrates the general shape for graphs of functions of the.

If we assume that b is non zero and that's a reasonable assumption because b to different powers are non zero, this is going to be zero for any non zero b. This is going to be zero right there, over here. We have the point one comma zero, so it's that point over there. Notice this point corresponds to this point, we have essentially swapped the x's and y's. In general when you're taking an inverse you're going to reflect over the line, y is equal to x and this is clearly reflection over that line.

Now let's look over here, when x is equal to four what is log base b of four. What is the power I need to raise b to to get to four. We see right over here, b to the first power is equal to four.

### 1. Definitions: Exponential and Logarithmic Functions

We already figured that out, when I take b to the first power is equal to four. This right over here is going to be equal to one. When x is equal to four, y is equal to one. Notice once again, it is a reflection over the line y is equal to x.

When x is equal to 16 then y is equal to log base b of The power I need to raise b to, to get to Well we already know, if we take b squared, we get to 16, so this is equal to two. When x is equal to 16, y is equal to two. Notice we essentially just swapped the x and y values for each of these points. This is y, this is a reflection over the line y is equal to x. Now, let's actually do that on the actual interface.

The whole reason is to give you this appreciation that these are inverse functions of each other. Let's plot the points.

That point corresponded to that point, so x zero, y one corresponds to x one, y zero. Here x is one, y is four that corresponds to x four, y one.

### The logarithm and exponential functions

Turn on complex numbers if you want to be able to evaluate the base 10 logarithm of a negative or complex number. Click the Simplify button. Algorithm for the base 10 logarithm function Click here to see the algorithm that computers use to evaluate the base 10 logarithm function.

The natural logarithm function Background: You might find it useful to read the previous section on the base 10 logarithm function before reading this section.

## 1. Definitions: Exponential and Logarithmic Functions

The two sections closely parallel each other. But why use base 10? After all, probably the only reason that the number 10 is important to humans is that they have 10 fingers with which they first learned to count.

Maybe on some other planet populated by 8-fingered beings they use base 8! In fact probably the most important number in all of mathematics click here to see why is the number 2. It will be important to be able to take any positive number, y, and express it as e raised to some power, x.

We can write this relationship in equation form: How do we know that this is the correct power of e?

Because we get it from the graph shown below. Then we plotted the values in the graph they are the red dots and drew a smooth curve through them. Here is the formal definition. The natural logarithm is the function that takes any positive number x as input and returns the exponent to which the base e must be raised to obtain x. It is denoted ln x. Evaluate ln e 4. The argument of the natural logarithm function is already expressed as e raised to an exponent, so the natural logarithm function simply returns the exponent.

Express the argument as e raised to the exponent 1 and return the exponent. Express the argument as e raised to the exponent 0 and return the exponent. The domain of the natural logarithm function is all positive real numbers and the range is all real numbers.

The natural logarithm function can be extended to the complex numbersin which case the domain is all complex numbers except zero. The natural logarithm of zero is always undefined. Then finding x requires solving this equation for x: The natural logarithm function is defined to do exactly the opposite, namely: How to use the natural logarithm function in the Algebra Coach Type ln x into the textbox, where x is the argument.

In floating point mode the natural logarithm of any number is evaluated. In exact mode the natural logarithm of an integer is not evaluated because to do so would result in an approximate number. Turn on complex numbers if you want to be able to evaluate the natural logarithm of a negative or complex number.

Algorithm for the natural logarithm function Click here to see the algorithm that computers use to evaluate the natural logarithm function. You might find it useful to read the previous section on the natural logarithm function before reading this section.

There we saw that it is possible to use the number e which is approximately 2. All three of these curves are called exponential functions because the independent variable x is in the exponent.