difference between exponents and powers

Question

I've heard that logarithms are the inverse of exponents and roots are the inverse of powers, but I've also heard that logarithms and roots are both inverses of exponents, in which case powers are not really defined. I think when it comes to functions, exponential functions have $x$ as the variable while polynomials are also called power functions since the exponents are numbers. Is this correct, or am I missing something?

Answer 1

The main difference is that exponential functions have a variable as the 'power', whereas in polynomial functions, the variable is the base of the power ($2^x$ vs $x^2$, for example).

Logarithms help with bringing the variable down from being a power to multiplying by a constant, so we can then use regular algebra to isolate it. Roots, on the other hand, tell us what number, multiplied by itself n times, gives us what we started with. (e.g. the cube root of $8$ gives us $2$, since $2$ multiplied by itself $3$ times gives us $8$.)

Hope this helps!