# Exponents with negative and positive bases in a relationship

### Positive and Negative Integer Exponents Multiplication. If two powers have the same base then we can multiply the powers . . A negative exponent is the same as the reciprocal of the positive exponent. Demonstrates how to use exponents on negative numbers. Points out The square means "multiplied against itself, with two copies of the base". This means . Home > Math > Topics > Exponents > Positive and Negative Integer Exponents I. Multiplying When you multiply same bases you add exponents. x4 •x5 = x4+5. The base 10 logarithm function can be extended to the complex numbersin which case the domain is all complex numbers except zero. The base 10 logarithm of zero is always undefined. Then finding x requires solving the following equation for x. But the original equation says that that exponent is x. Note that if c is negative then there is no real solution. However there is a complex solution. The base 10 logarithm function is defined to do exactly the opposite, namely: Therefore these are inverse functions.

We should look at these two equations as expressing the same relationship between x and y but from different points of view. The first equation is the relationship solved for y and the second one is the relationship solved for x. If we replace x in the first equation by the x of the second equation we get this identity: This always happens with inverse functions.

How to use the base 10 logarithm function in the Algebra Coach Type log x into the textbox, where x is the argument.

The argument must be enclosed in brackets. Set the relevant options: In floating point mode the base 10 logarithm of any number is evaluated.

In exact mode the base 10 logarithm of an integer is not evaluated because doing so would result in an approximate number.

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.

The two sections closely parallel each other. But why use base 10?

### Exponents with negative bases (video) | Khan Academy

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.

Whenever we raised raised a negative base to an exponent, if we raise it to an odd exponent, we are going to get a negative value. And that's because when you multiply negative numbers an even number of times, a negative number times a negative number is a positive.

## Powers and exponents

But then you have one more negative number to multiply the result by — which makes it negative. And if you take a negative base, and you raise it to an even power, that's because if you multiply a negative times a negative, you're going to get a positive. And so when you do it an even number of times, doing it a multiple-of-two number of times. So the negatives and the negatives all cancel out, I guess you could say.

Or when you take the product of the two negatives, you keep getting positives. So this right over here is going to give you a positive value. So there's really nothing new about taking powers of negative numbers. It's really the same idea. And you just really have to remember that a negative times a negative is a positive. And a negative times a positive is a negative, which we already learned from multiplying negative numbers.

## Negative exponents

Now there's one other thing that I want to clarify — because sometimes there might be ambiguity if someone writes this. Let's say someone writes that. And I encourage you to actually pause the video and think about with this right over here would evaluate to. And, if you given a go at that, think about whether this should mean something different then that.

Well this one can be a little bit and big ambiguous and if people are strict about order of operations, you should really be thinking about the exponent before you multiply by this So many times, this will usually be interpreted as negative 2 to the third power, which is equal to -8, while this is going to be interpreted as -2 to the third power.

Now that also is equal to You might say well what's what's the big deal here? Well what if this was what if these were even exponents. So what if someone had give myself some more space here. What if someone had these to express its -4 or a -4 squared or -4 squared.