what is a logarithmic function?

Question

I never understand the difference between logarithmic and exponential functions. Why is an exponential function of the form $y=2^x$ but a logarithmic function of the form $y=\log_2{x}$ ?

Answer 1

Exponential and logarithm are, more or less, "inverse" functions of one another just like $+$ and $-$ are inverse functions of one another.

As an example you can say that $4=2^2$ as well as saying $2=\log_2 4$. The logarithm is to be red as follows

Given three numbers $a,b,c\in\mathbb{R}$ such that $$c = \log_a b$$ then $c$ is the exponent to give to $a$ to get as a result $b$, which in mathematical form is the same as writing $$a^c = b$$

To expand on the "inverse" meaning, suppose that we want to find the solution to the equation $$x+5 = 0$$then you just have to put a $\color{red}{-}5$ to both sides and get the solution. Now suppose that you want to find the solution to $$5x = 1 $$ then you just divide by $5$ both sides and get your solution. Morover suppose that you want to find the solution to the equation $$2^x=4$$ then you just take the $\log_2$ of both sides $$\log_2(2^x)=\log_2 4 \implies x = 2 $$

Answer 2

Exponential function : If $~b~$ be any number such that $~b\gt 0~$ and $~b\neq 1~$ then an exponential function is a function in the form,$$y(x)=~b^x$$ where $~b~$ is called the base , the exponent,$~x~$ can be any real number.

In mathematics, the natural exponential function is $~y(x)=e^x~,$ where $e$ is Euler's number. It is a special exponential function. In fact this is so special that for many people this is THE exponential function.

• When $~b=2~$, then $~y=2^x~$ is an exponential function with base $2$.

---

Logarithmic function : If $~b~$ be any number such that $~b\gt 0~$ and $~b\ne 1~$ and $~x\gt 0~$ then, $$y(x)=\log_bx$$is called the logarithmic function.

• $~y(x)=\log_bx\equiv b^y=x~$.
• When $~b=2~$, then $~y=\log_2x~$ is a logarithmic function with base $2$.

Notes:

• The two most common bases that we use are base $~10~$ and base $~e~$. When we use base $~e~$, then we use any one of the form either $~\log_ex~$ or $~\ln x~$.

• A logarithm is simply an exponent that is written in a special way.

• The inverse of an exponential function is a logarithmic function and the inverse of a logarithmic function is an exponential function.

---

---

Difference between exponential function and logarithmic function :

• The exponential function is given by $~ƒ(x) = e^x~$, whereas the logarithmic function is given by $~g(x) = \ln x~$, and former is the inverse of the latter.

• The domain of the exponential function is a set of real numbers, but the domain of the logarithmic function is a set of positive real numbers.

• The range of the exponential function is a set of positive real numbers, but the range of the logarithmic function is a set of real numbers.

• Observations from the graphs :

The graph of the exponential function $~ƒ(x) = e^x~$ never crosses the $x$-axis $($it is because there is no value of $x$ that will cause the value of $f(x)$ in the formula $~ƒ(x) = e^x~$ to equal $0)$. Also the graph crosses the $y$-axis at $1$. Because the value of $x$ is always zero on the $y$-axis. Substitute $0$ for $x$ in the equation $~ƒ(x) = e^x\implies ƒ(0) = e^0=1 ~$ . This translates to the point $(0, 1)$. On the graph, as the value of $x$ increases, the value of $f(x)$ also increases. This means that the function is an increasing function. As $x$ gets larger and larger, the function value of $f(x)$ is increasing more and more dramatically. This is why the function is called an exponential function.

The graph of the logarithmic function $~g(x) = \ln x~$ is located entirely in quadrants $\text I$ and $\text{IV}$ and never touches the $y$-axis. It means that the value of $x $ (domain of the function $g(x)$) in the equation $~g(x) = \ln x~$ is always positive. Because the equation $~g(x) = \ln x~$ can be rewritten as the exponential function $~x=e^{g(x)}~$ and hence there is no value of $g(x)$ that can cause the value of $x$ to be negative or zero. Also the graph of $~g(x) = \ln x~$ will never cross the $y$-axis because $x$ can never equal $0$. The graph will always cross the $x$-axis at $1$. As $x$ increases, the value of $g(x)$ is also increases. This means that the function is an increasing function. Clearly, the increase in the value of the function is most dramatic between $0$ and $1$. After $x = 1$, as $x$ gets larger and larger, the increasing function values begin to slow down (the increase get smaller and smaller as $x$ gets larger and larger). Again the function values are positive for $x$'s that are greater than $1$ and negative for $x$'s less than $1$.