Algebra
Textbooks
Boundless Algebra
Exponents, Logarithms, and Inverse Functions
Introduction to Exponents and Logarithms
Algebra Textbooks Boundless Algebra Exponents, Logarithms, and Inverse Functions Introduction to Exponents and Logarithms
Algebra Textbooks Boundless Algebra Exponents, Logarithms, and Inverse Functions
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 10
Created by Boundless

Introduction to Exponential and Logarithmic Functions

Logarithmic functions and exponential functions are inverses of each other. That is, they undo each other. 

Learning Objective

  • Explain the relationship between logarithmic functions and exponential functions


Key Points

    • An exponent of $-1$ denotes the inverse function. That is, $f^{-1}(x) $ is the inverse of the function $f(x)$.
    • An inverse function is a function that undoes another function: If an input $x$ into the function $f$ produces an output $y$, then inputting $y$ into the inverse function $g$ produces the output $x$, and vice versa (i.e., $f(x)=y$, and $g(y)=x$). 
    • The logarithm to base $b$ is the inverse function of $f(x) = b^x$: $\log _{ b }{ { (b) }^{ x } } = x\log _{ b }{ (b) } =x$
    • The natural logarithm $ln(x)$ is the inverse of the exponential function $e^x$:$b={ e }^{ lnb }$

Term

  • inverse function

    A function that does exactly the opposite of another.


Full Text

The inverse of an exponential function is a logarithmic function and vice versa. That is, the two functions undo each other. Thus $log_{b}b^{x}=x $ and $b^{log_{b}x}=x $. Composing the functions in either order leaves the initial input unchanged. 

Another way of saying this is that if $f(x)=log_{b}x $, then the inverse function is given by $f^{-1}(x)=b^{x} $, and vice versa.

Lastly, as with all inverse functions, if we graph $f(x)=log_{b}x $  and $f^{-1}(x)=b^{x}$ on the same plane, the graphs will be symmetric across the line $y=x$. That is, if we fold the plane over the line $y=x$, the two curves will lie on each other. Another way of thinking about this is that if we generate points on the curve of $f(x)=log_{b}x$ we can find the points on the curve of  $f^{-1}(x)=b^{x}$ by interchanging the $x$ and $y$ coordinates of the points.

In the following graph you can see an exponential function in red and its inverse, a logarithmic function, in blue. The graphs are symmetric over the line $y=x$, which is pictured in black. Further, a point $(t,u=b^t)$ on the graph of $f(x)$ yields a point $(u,t=log{_b}u)$ on the graph of the logarithm and vice versa. 

Logarithm function

The graph of the logarithm function $log_b(x)$ (blue) is obtained by reflecting the graph of the function $b(x)$ (red) at the diagonal line ($x=y$).

Thus far we have been looking at logs of the base $b$. Let us consider instead the natural log (a logarithm of the base $e$).  The natural logarithm is the inverse of the exponential function $f(x)=e^x$. It is defined for $e>0$, and satisfies $f^{-1}(x)=lnx $.

As they are inverses composing these two functions in either order yields the original input. That is, $e^{lnx}=lne^x=x$.

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