Logarithmic Functions

The basic form of a logarithmic function is y = f(x) = log_bx (0 < b ≠ 1), which is the inverse of the exponential function b^y = x. 

The logarithmic functions can be in the form of ‘base-e-logarithm’ (natural logarithm, ‘ln’) or ‘base-10-logarithm’ (common logarithm, ‘log’).

Here are some examples of logarithmic functions:

• f(x) = log₅x
• g(x) = log(3x – 1)
• h(x) = ln(4x) + 2

Finding Domain and Range

The domain of the function y = log_bx is x > 0 or (0, ∞) and the range of any logarithmic function is the set of real numbers.

Let us determine the domain of the logarithmic function g(x) = log(3x – 1)

Setting the argument, 3x – 1 > 0, we get

3x > 1

⇒ x > ${\dfrac{1}{3}}$

Thus, the domain is ${\left( \dfrac{1}{3},\infty \right)}$ and the range is the set of ℝ.

Finding Asymptote

As we know, the domain of the above function g(x) = log(3x – 1) is ${\left( \dfrac{1}{3},\infty \right)}$.

Now, to find the asymptote of the logarithmic function, we solve for x by putting 3x – 1 = 0.

We get 3x = 1

⇒ x = ${\dfrac{1}{3}}$

Thus, the vertical asymptote is x = ${\dfrac{1}{3}}$. 

Graphing Basic Logarithmic Functions

Let us graph the logarithmic function y = f(x) = log₅x considering its inverse (in the form of the exponential function) 5^x = y

Now, the exponential function y = 5^x can be graphed as

The graph representing the inverse of a function mirrors the graph of the original function.

Thus, the logarithmic function y = log₅x, the inverse of y = 5^x, is the reflection of the previous graph, as shown.

Alternative Method

However, the graph of the logarithmic function with base ‘b’ (y = log_bx) can also be obtained by converting its exponential form x = b^y 

Let us plot y = log₃x.

First, we convert y = log₃x to its exponential form, x = 3^y

Now, considering the values of x and y, we get:

y	x = 3y	(x, y)
-2	${3^{-2}=\dfrac{1}{9}}$	${\left( \dfrac{1}{9},-2\right)}$
-1	${3^{-1}=\dfrac{1}{3}}$	${\left( \dfrac{1}{3},-1\right)}$
0	30 = 1	(1, 0)
1	31 = 3	(3, 1)
2	32 = 9	(9, 2)

By plotting all the points to graph the function y = log₃x, we get

By summarizing all these methods, the graphs of the given logarithmic functions are represented as follows: 

We observe that the graph y = log_bx has domain D = {x: x > 0}, a vertical asymptote at x = 0, and an x-intercept at x = 1. Also, if b > 1, the graph increases, and if 0 < b < 1, it decreases, and the range is all the real numbers.

Transformations of Graphs

Transformations of any logarithmic function’s graph are similar to those of the other function, including shift, stretch, compress, and reflection of the parent function y = f(x) = log_bx.

Vertical Shifts

When we add or subtract a constant ‘c’ to the logarithmic function f(x) = logbx, we get a result that is vertically shifted in the direction of the sign on ‘c.’ 

Here, the curve of f(x) = log_bx is formed alongside the shift up (g(x) = log_bx + c) and the shift down (h(x) = log_bx – c), as shown.

Here, the graph of f(x) = log_b(4x) + 2 along with its parent function are shown:

Horizontal Shifts

The logarithmic function f(x) = log_bx can be shifted ‘d’ units horizontally with respect to the equation f(x) = log_b(x + d). Thus, if d > 0, the curve f(x) = log_b(x + d) shifts in the left direction, whereas when d < 0, the curve f(x) = log_b(x – d) shifts in the right direction.

The graph of f(x) = log_bx with its shifted forms is shown.

Reflections

When we multiply the parent function f(x) = logbx by (-1), we get the result as a reflection on the x-axis. Similarly, when we multiply the argument of the function f(x) = log_bx with (-1), we get the reflection on the y-axis.

Here is a general curve of the logarithmic function f(x) = log_bx with its reflections of the x-axis (g(x) = – log_bx) and y-axis (h(x) = log_b(-x)). 

For example, the graph of g(x) = ln(-x), along with its parent function f(x) = ln(x) is shown:

Vertical Stretches and Compressions

On multiplying the parent function f(x) = log_bx by the constant ‘m’ (m > 0), we get a vertical stretch or compression of the original graph, and the functions are:

• Vertical stretch, g(x) = m log_b(x)
• Vertical compression, h(x) = ${\dfrac{1}{m}\log _{b}\left( x\right)}$

Horizontal Stretches and Compressions

On multiplying the argument of the parent function f(x) = log_bx by the constant ‘n’ (n > 0), we get the horizontal stretch or horizontal compression of the original graph, and the functions are:

• Horizontal stretch, g(x) = log_b(nx), here n > 0
• Horizontal compression, h(x) = log_b(nx), here 0 < n < 1

Here are the graphs of the functions log(x), log(4x), and log(18x).

Finding the Equation of a Graph

Now, let us find a possible equation for the following graph of a common logarithmic function.

Since the given logarithmic function is horizontally shifted to the left and has a vertical asymptote at x = -2, the argument of the logarithmic function is (x + 2). 

Again, since the graph is vertically shifted down and reflected on the y-axis, the logarithmic equation can be written as y = m log_b(x + 2) – c.

As we know, log_b(1) = 0.

Thus, if x + 2 = 1 (or, x = -1), y = -c. 

From the graph, when x = -1, we get y = -7, which implies -c = -7

Now, the logarithmic equation is y = m log_b(x + 2) – 7

Also, log_b(b) = 1

Thus, if x + 2 = b, y = m – 7. 

Since the given graph is of a common logarithmic function with base 10 (b = 10)

If b = 10, x = b – 2 = 8. 

From the graph, when x = 8, we get y = 3, and m = y + 7 = 10

Thus, the logarithmic equation of the given graph is y = 10 log₁₀(x + 2) – 7 or 10 log(x + 2) – 7

Alternative Method

As we know, the general form of the logarithmic function is y = m log_b(x + 2) – c, and the graph passes through the points (-1, -7) and (8, 3).

Substituting the values x = -1 and y = -7, we get 

-7 = m log_b(-1 + 2) – c

⇒ -7 = m log_b(1) – c

⇒ -7 = -c (since log_b(1) = 0)

Now, the equation is y = m log_b(x + 2) – 7

By substituting the values x = 8 and y = 3, we get

3 = m log_b(8 + 2) – 7

⇒ 3 + 7 = m log_b(10)

⇒ 10 = m log_b(10)

⇒ m = ${\dfrac{10}{\log _{b}\left( 10\right) }}$

Thus, the equation is:

${y=\dfrac{10}{\log _{b}\left( 10\right) }\log _{b}\left( x+2\right) -7}$

⇒ ${y=10\dfrac{\log _{b}\left( x+2\right) }{\log _{b}\left( 10\right) }-7}$

⇒ ${y=10\log _{10}\left( x+2\right) -7}$ (by the change of base rule)

Thus, the logarithmic equation of the given common logarithmic graph is y = 10 log₁₀(x + 2) – 7 (or, 10 log(x + 2) – 7)

Solved Examples