Question

Explain how we know that logarithmic functions are one to-one. Why does this mean that $A=B$if and only if ${\log }_{b}A={\log }_{b}B$ (assuming that $A$ and $B$ are positive)?

Short answer

The logarithmic functions are one to one and it means that$A=B$if and only ifrole="math" localid="1648996944804" ${\log }_{b}A={\log }_{b}B.$

Step-by-step solution

Step 1. Given information

We need to explain how the logarithmic functions are one to one and why does it means that$A=B$if and only if${\log }_{b}A={\log }_{b}B.$(Assuming that$A$andare positive).

Step 2. Explanation

Consider the natural logarithmic it is the inverse of an exponential function. So if an exponential function is one to one then logarithmic functions are also.

A one-to-one is a function $f\left(x\right)$that each element of $f\left(x\right)$has a unique element in its domain.

Thus if $A=B.$Taking log with base $b$in both sides.

${\log }_{b}A={\log }_{b}B.$

Conversely if ${\log }_{b}A={\log }_{b}B$.

Remove log with the base $b$onboth sides.

$A=B.$