Exponential vs Logarithmic Functions

Two functions built from the same relationship read in opposite directions. Converting between the power form and the logarithm form, and just how far apart their growth rates are.

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Exponential and logarithmic functions are built from the same relationship read in opposite directions. One asks what is the result?, the other asks what is the exponent? — and the difference in how fast they grow is enormous.

Concept The basic comparison
f(x) = b^x — asks what the result is. Grows very quickly.
g(x) = \log_b(x) — asks what the exponent is. Grows very slowly.

Each is the inverse of the other: if b^3 = 8 , then \log_b(8) = 3 . The same three numbers, arranged differently.

Concept The inverse relationship

b^a = c \quad \Longleftrightarrow \quad \log_b(c) = a

These two statements carry identical information. Whenever one is true the other is too, which is why converting between them is often the first step in solving a problem.

Example The difference in growth rate

Compare 2^x and \log_2(x) at the same inputs:

x = 1 : 2^1 = 2 , while \log_2(1) = 0 .
x = 2 : 2^2 = 4 , while \log_2(2) = 1 .
x = 4 : 2^4 = 16 , while \log_2(4) = 2 .
x = 16 : 2^{16} = 65{,}536 , while \log_2(16) = 4 .
⟹ at x = 16 one gives 65,536 and the other gives just 4

That gap is the whole point. Exponential growth runs away; logarithmic growth barely moves — which is exactly why logarithms are used to compress very large ranges of numbers.

Note The two curves
bˣ log
The dashed line is y = x . Fold the page along it and the two curves land on each other — the visual signature of inverse functions.
Example Converting between the two forms

Rewrite 5^3 = 125 as a logarithm, and \log_4(64) = 3 as a power.

In 5^3 = 125 , the base is 5, the exponent 3, the result 125.
So \log_5(125) = 3 .
In \log_4(64) = 3 , the base is 4, the result 64, the exponent 3.
So 4^3 = 64 .
⟹ log₅(125) = 3 and 4³ = 64

The base stays the base in both forms. Only the positions of the exponent and the result change places.

Note Mistakes to avoid
Moving the base when converting between forms — it never changes position.
Expecting logarithmic growth to keep pace with exponential growth.
Confusing which function takes the exponent as input and which returns it.
Forgetting that a logarithm's input must be positive, while an exponential accepts any x .
Assuming the curves cross the same axes; one meets (0,1) , the other (1,0) .
Summary
  1. The exponential asks for the result; the logarithm asks for the exponent.
  2. b^a = c and log_b(c) = a say exactly the same thing.
  3. Exponential growth is extremely fast; logarithmic growth is extremely slow.
  4. At x = 16 with base 2, one gives 65,536 while the other gives 4.
  5. The graphs are reflections of each other in the line y = x.
231. Arithmetic Sequences and Series
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