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# Logarithmic Functions: math.log()

**00:00**
Usually, after you study exponential functions, you move on to logarithmic functions, and this is what we’ll do in this lesson. In particular, we’ll take a look at the logarithm function defined in the `math`

module.

**00:13**
Okay, so you made it this far. You got through the radioactive decay problem, and so now you’re in for a treat because we’re going to take a look at logarithmic functions. I know, I know, settle down. Let’s get through this.

**00:26**
So, here’s how these functions come up. Say you have an exponential function 2.5 to the power of *x*, and you want to know for what value of input *x* will—when raising 2.5 to the power of that *x*—are you going to get 39.0625? If you think about it, this is an inverse problem.

**00:46**
You want to know “What *x* gives me this output?” This is where logarithmic functions come in. Every exponential function *a* to the *x* has an inverse.

**00:58**
The inverse of *a* to the *x* is the logarithm function with base *a*. These functions are denoted by log subscript *a*, the input is *x*.

**01:07**
The constant is *a* here—it’s fixed—and the thing that varies is *x*. So, what can you do with this logarithm function base *a*? Well, you can solve this inverse problem.

**01:20**
The logarithm with base 2.5 evaluated at the input 39.0625, we get 4. So that means that 2.5 raised to the power of 4 is 39.0625. So, using the logarithm with base 2.5 solves our inverse problem.

**01:45**
So, in Python, to compute the logarithm base *a* evaluated at *x*, we use the `log()`

function. It takes two inputs. The first one is `x`

and the second one is the base.

**01:58**
Both of these values `x`

and `a`

, they must be positive numbers.

**02:04**
Okay, one last thing before we get on to testing these functions. There are two other frequently used logarithm functions besides the natural logarithm. This is when *a* is equal to 2, so instead of calling `math.log(x, 2)`

, there’s the dedicated function `log2()`

in the `math`

module.

**02:24**
This is going to be more accurate than using `log()`

with `x, 2`

. And then the other one is when the base is *a* = 10, and instead of writing `log(x, 10)`

, there’s the dedicated function in the `math`

module called `log10()`

. And again, this one’s going to be more accurate than calling `log()`

with `x`

and then with a `10`

for the base. So, now let’s test these functions in Python.

**02:52**
Let’s compute a couple of values with the logarithm function. The logarithm of `54`

, say, base `3`

.

**03:02**
As you saw, the logarithm functions—they are the inverse functions of the corresponding exponential functions. Let’s save this value in the variable `x`

, and we know that the logarithm with base `3`

is the inverse function of the exponential function with base `3`

. Now, if we computed that the logarithm of 54 base 3 is 3.63 and so on, and we’re saving that in the variable `x`

, that means then that `3`

raised to the power of `x`

should be `54`

. So, modulo some rounding error, we see that this is working well.

**03:41**
Logarithm functions are the inverses of the corresponding exponential functions.

**03:47**
Every exponential function—it doesn’t matter what the base is, you’re always going to get a positive number. That means that the only numbers you can input into the logarithm as the first argument, they must be positive. For example, if you try the logarithm , say, `-54`

base `3`

, what you’re asking for here is “If the base is 3 and I raise 3 to the power of some number, what gives me -54?” and that’s what this would compute. But you get a domain error, a `math domain error`

, and that’s because there’s no number that you can raise 3 to the power of to get -54. You’re always going to get a positive number.

**04:29**
Let’s check out the couple of built-in special logarithm functions. These were `math.log2()`

, so this will compute the logarithm of whatever input with base 2.

**04:40**
So, for example, `32`

—that’s `5`

. So `2`

to the power of `5`

is `32`

.

**04:50**
Now go ahead and try, say, using `log10()`

. That was the other special log function. Just for convenience, Python defines these two because they’re frequently used.

**04:59**
Go ahead and try, say, the logarithm of `1000`

base `10`

. So, what number do you need to raise 10 to the power of to get 1,000? That’s 3.

**05:09**
Now let’s compare the accuracy if instead of using `log10()`

, we used, say, `log()`

with the second argument passing in the base.

**05:19**
So in this case, we’re off a little bit again due to rounding, and so this is a good example of why you would want to use these specialized functions `log2()`

and `log10()`

.

**05:28**
They’re going to be more accurate than if you were to use the `log()`

function with the corresponding base. All right, so that is the basic usage of the logarithm function and the two specialized ones `log2()`

and `log10()`

. In the next lesson, we’ll take a look at the natural logarithm in an application using the decay problem with Californium-252.

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