I'm currently studying the properties of logarithm in an open source pre-calculus textbook that can be found here (Page 438). Before the text goes on to the Algebraic properties of exponential and logarithmic functions it defines the "one-to-one" properties of exponential function and logarithmic functions (Theorem 6.4). As you can see:
And on the next page it describes the algebraic properties of logarithms...
Is there any particular reason for stating the one-to-one property of logarithmic and exponential function beforehand? I know the logarithmic function is the inverse of an exponential function, and as such it can only exists for a one-to-one function, but is there any other reason why this was mentioned? I have the feeling that this distinction is extremely important with regards to the algebraic properties of logarithms that come after (i.e. product rule, quotient rule, power rule) but I cannot see why.
I'm sorry if this was long winded, and thank you for taking the time to read this. For who ever is so kind as to answer could you bear in mind that I'm studying at a pre-calculus level. Thanks.