Logarithms?

Question

Compare f(x) = a^x and f(x) = log[base a]x in terms of:

1. Intercepts (x and/or y)
2. Symmetry (to each other)
3. The concept of inverse (including f of f^-1(x))

Any help would be appreciated.

I solved most of it.

1.
For f(x) = log[base a]x
For any base, the x-intercept is 1, because the logarithm of 1 is 0.
Proper fractions will have negative logarithms.
Logarithmic functions are only defined for positive values of x.
The range is all real numbers.

For f(x) = a^x
Defined for every real number x
The y-intercept will always be at 1, because a^0=1
Range is the collection of all positive real numbers

2. Reflection on line y=x

3. For any base a, the two functions are inverses.
f(f^-1(x)) = x for any x in the doman of f(x).

Answer 1

You are right.
1. Since by definition y = log[a]x means x = a^y those two functions are inverse of each other therefore symmetric around y = x line.
2. Since a^0 = 1 the first function intercepts axis y at y = 1. Similarly y = log[a]x intercepts x at x = 1.
3. For a <= 1 the two functions intercept each other at x = a^x.

Answer 2

hint: opposite powers through using logarithms. So opposite a^n through taking the log of a^n with a base a; n=log[a](a^n) And opposite log base a through taking the potential of a. a^(log[a](n))=n.