# Thread: Base is a number, exponent is a variable

1. ## Base is a number, exponent is a variable

Is $\displaystyle 3^z$ a constant or a variable?

2. Originally Posted by lfroehli
Is $\displaystyle 3^z$ a constant or a variable?
It's a variable.

3. Thanks! Can you explain why, please?

4. Because it represents some number, but which number in particular is DEPENDENT on the variable in the expression, z.

EDIT: By definition, a constant is a number that is constant; that is, unchanging.

5. If z= 0, $\displaystyle 3^z= 3^0= 1$. If z= 1, $\displaystyle 3^z= 3^1= 3$.

It varies therefore it is variable!

6. Hopefully others can supplement the following with more precise language...

3^(something) could be interpreted in a few ways, depending on what that "something" is.

If "something" is a number, then it's obvious: 3^2 = 3^2(!) = 9.
If "something" is, say, a/b/c/d, then we might interpret a/b/c/d to be a constant, in which case 3^a, while still "depending" on the value of a, doesn't vary once we have defined a.
If "something" is, say, u/x/y/z, then - as everyone has to this point - we interpret these letters are being variable.

Thoughts?

Yeah, I know. It's wikipedia. But this way, we're all pulling from the same definition!

7. In algebraic functions such as f(x) = x^2 g(x)=x^5 the base is variable and the exponent is constant
In exponential functions the base is constant and the exponent is variable
f(x)=2^x g(x)=5^x

bjh

8. The reason I ask is because I am trying to find the derivative of $\displaystyle 3^z$. If it is a constant or number, the derivative would be 0. If it's not a number, then I don't know what it would be because you can't use the power rule.

9. We will assume that z is a variable (by convention - see my previous post).

The derivative of $\displaystyle a^z$ is $\displaystyle a^z*ln(a)*z'$

This is an exponential function with base 3. Notice that we use the chain rule (hence the z' in the derivative) and when the base is "e", then you just get

$\displaystyle e^z*ln(e)z' =$

$\displaystyle e^z*1*z' =$

$\displaystyle e^zz'$

10. $\displaystyle 3^z$ is a variable expression.

11. Originally Posted by skeeter
$\displaystyle 3^z$ is a variable expression.
For whose benefit is this post? I feel like there are four conversations going on, yet nobody (else) is addressing the OP's needs...

12. I was addressing the OP's question ...

Originally Posted by lfroehli
Is $\displaystyle 3^z$ a constant or a variable?