definition or property of logarithms

Question

I've seen a lot of complicated logarithm definitions on this StackExchange and I have a rather simple question:

$$a^{b}=c \leftrightarrow \log_a{c}=b$$

Is this a definition of logarithms, which all other properties would follow from? Or is this simply a property that follows from other definitions?

Answer 1

Your equation is precisely the definition of the logarithm.

The logarithmic identities that follow from this definition (and from the properties of exponentiation) are:

• $\log_a(x)+\log_a(y) \equiv\log_a(xy)$ (since $a^\alpha \cdot a^\beta \equiv a^{\alpha+\beta}$)

• $\log_a(x^n) \equiv n \log_a(x)$ (since $[a^{\alpha}]^\beta \equiv a^{\alpha \cdot\beta}$)

• $\log_a(x)-\log_a(y) \equiv\log_a\left( \frac{x}{y} \right)$ (Why? Exercise!)

• $\log_a(1) \equiv0$ (since $a^0 \equiv 1$)

• $\log_a(a)\equiv1$ (Why? Exercise!).

Answer 2

It is the definition of logarithm. You can use this definition to derive the other properties of logs.