Logarithmic Function and Its Properties: In Mathematics, many scholars use logarithms to change multiplication and division questions into addition and subtraction questions. In logarithms, the power is raised to a number to get a different number. It is an inverse function of the exponential function, and an exponential function can be expressed as a logarithmic function.

We know, Mathematics and Science constantly deal with the large exponents of numbers. Hence, logarithms are important and useful in various fields. In this article, let us learn about logarithmic functions, rules, properties, and some examples.

Learn Detailed Properties of Logarithm

Definition of Logarithmic Function

The logarithmic function is an inverse function of an exponential function, so the logarithmic function is defined as

For \(x > 0,\,a > 0\) and \(a \ne 1\)

 \(y = {\log _a}x\) if and only if \(x = {a^y}\), then the function is given by

\(f(x) = {\log _a}x\)

Here,

\(a \to \) base or logarithm

This can be read as log base \(a\) of \(x\).

Types of Logarithms

Logarithms can be divided into two types

Common Logarithms

• The logarithms of the base \(10\) are called common logarithms

Example: \({\log_{10}}25,\,{\log_{10}}10,\,{\log _{10}}16\)

Natural Logarithms

• The logarithms of the base \(e\) are called natural logarithms

Example: \({\log _e}10,\,{\log _e}400\)

In this article, let us learn more about common logarithms.

Finding the value of \(x\) in the exponential expressions such as \({2^x} = 4,\,{2^x} = 16\) is easy, but in \({2^x} = 12\), is somewhat tricky to find. So to find the value of \(x\) we can use logarithmic functions and transform \({2^x} = 12\) into a logarithmic form as \({\log _2}12 = x\) and then find the value of \(x\).

Hence, the logarithmic form of \({2^x} = 12\) is \({\log _2}12 = x\).

Mathematically, logarithmic functions are inverses of exponent functions.

Practice Questions on Logarithm and Its Application

Graph of Logarithmic Functions

 If \(a > 0\) and \(a \ne 1\). Then the function defined by \(f(x) = {\log _a}x,\,x > 0\) is called the logarithmic function.

Note: Functions \(f(x) = {\log _a}x\) and \(g(x) = {a^x}\) are inverse each other. So their graphs are mirror images of each other in the mirror line \(y = x\).

Observe that the logarithmic and the exponential functions are inverses of each other.

\( \Rightarrow {\log _a}x = y \Leftrightarrow x = {a^y}\)

Rules of Logarithmic Functions

The basic rules for the logarithms are listed below.

Rule 1: Product rule

Multiply two numbers with the same base, then add exponents

\({\log _b}(MN) = {\log _b}M + {\log _b}N\)

Rule 2: Quotient rule

Divide the two numbers with the same base, then add the exponents

\({\log _b}\left( {\frac{M}{N}} \right) = {\log _b}M – {\log _b}N\)

Rule 3: Power rule

Raise an exponential expression to power and multiply the exponents.

\({\log _b}\left( {{M^k}} \right) = k \cdot {\log _b}M\)

Rule 4: Zero rule

\({\log _b}(1) = 0\)

Rule 5: Identity rule

\({\log _b}(b) = 1\)

Rule 6: Log of exponent rule

\({\log _b}\left( {{b^k}} \right) = k\)

Rule 7: Exponent of log rule

\({b^{{{\log }_b}(k)}} = k\)

Here, \(b > 0\) but \(b \ne 1\), and \(M,\,N\), and \(k\) are real numbers, but \(M,\,N\) must be positive numbers.

Plotting Graphs

For example, the exponential function \(f(x) = {2^x}\) gives the following table.

\(x\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(f(x)\)	\({\frac{1}{4}}\)	\({\frac{1}{2}}\)	\(1\)	\(2\)	\(4\)

Since a logarithmic function is the inverse of an exponential function, then \(g(x) = {\log _2}(x)\) gives the table as shown below.

\(x\)	\({\frac{1}{4}}\)	\({\frac{1}{2}}\)	\(1\)	\(2\)	\(4\)
\(g(x)\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)

Here, from first table, we observe that,

• As the input increases, the output increases.
• As the input increases further, the output increases slowly.

Since an exponential function only gives positive outputs, the logarithm function accepts only positive inputs.

• So, the domain of the logarithmic function is \((0,\,\infty )\).

Since the exponential function accepts all real numbers as inputs, the logarithm can give real numbers as output.

• The range of logarithmic functions is all real numbers, \(( – \infty ,\,\infty )\).

Plotting the graph of \(g(x) = {\log _2}(x)\) from the points in the table, notice that as the input values for \(x\) approach zero, the output grows in the negative direction, indicating a vertical asymptote at \(x = 0\).

Note: The graph of a logarithmic function has a vertical asymptote at \(x = 0\). Recall that an asymptote is a line whose distance from a curve approaches zero as the coordinates tend to infinity.

Graphs of logarithmic functions decrease from left to right, if \({\rm{0 < }}a{\rm{ < 1}}\). If the base of the logarithmic functions, \(a > 1\), then the graph will increase from left to right.

We can write

• As \(x \to {0^ + },\,f(x) \to – \infty \) and \(x \to \infty ,\,f(x) \to \infty \)
• As \(a > 0\) and \(a \ne 1\)

Now, we have the following cases.

Case (i): \(a > 0\)

Here, we have

\(y = {\log _a}x\left\{ {\begin{array}{*{20}{c}}{ < 0}&{{\rm{ for }}\,0 < x < 1}\\{ = 0}&{{\rm{ for}}\,{\rm{ }}x = 1}\\{ > 0}&{{\rm{ for }}\,x > 1}\end{array}} \right.\)

The graph of \(y = {\log _a}x\) is as shown below.

Notice that the values of \(y\) increase with the increase in \(x\).

Case (ii): \({\rm{0 < }}a{\rm{ < 1}}\)

Here, we have

\(y = {\log _a}x\left\{ {\begin{array}{*{20}{c}}{ > 0}&{{\rm{ for }}\,0 < x < 1}\\{ = 0}&{{\rm{ for }}\,x = 1}\\{ < 0}&{{\rm{ for }}\,x > 1}\end{array}} \right.\)

Notice that the values of \(y\) decrease with the increase in \(x\).

So, the graph of \(y = {\log _a}x\) is as shown below.

Properties of the Logarithmic Graph

• The domain is \((0,\,\infty )\)
• The range of \(( – \infty ,\,\infty )\)
• The logarithmic function has a vertical asymptote at \(x = 0\)
• The \(x\)-intercept is \((1,\,0)\)
• There is no \(y\)-intercept, as the value \(f\left( 0 \right)\) is not defined since the domain of the function is \((0,\,\infty )\).
• It is either strictly increasing or strictly decreasing.

Logarithmic Function Properties

The following are some useful properties of logarithmic function:

Property Name	Property Expression
Logarithm of \(1\)	\({\log _a}1 = 0\)
Logarithm of the same number as base	\({\log _a}a = 1\)
Product rule	\({\log _a}(mn) = {\log _a}m + {\log _a}n\)
Quotient rule	\({\log _a}\left( {\frac{m}{n}} \right) = {\log _a}m – {\log _a}n\)
Power rule	\({\log _a}{m^n} = n{\log _a}m\)
Change of base rule	\({\log _b}a = \frac{{{{\log }_c}a}}{{{{\log }_c}b}}\) \({\log _b}a \cdot {\log _c}b = {\log _c}a\)
Equality rule	\({\log _b}a = {\log _b}c\) \( \Rightarrow a = c\)
Number raised to the logarithm	\({a^{{{\log }_a}x}} = x\)
Negative of a logarithm	\( – {\log _b}a = {\log _b}\frac{1}{a}\) \( – {\log _b}a = {\log _{\frac{1}{b}}}a\)
Reciprocal rule	\({\log _b}a = \frac{1}{{{{\log }_a}b}}\)

Solved Examples – Logarithmic Function and Its Properties

Some example problems are solved below to give a clear idea of Logarithmic Function and Its Properties.

Q.1. Express \({2^5} = 32\) in logarithmic form.
Ans: As we know, the exponential form \({a^x} = N\) can be written in logarithmic function as \({\log _a}N = x\)
Hence, \({2^5} = 32\) can be written in a logarithmic form as
\({\log _2}32 = 5\)

Q.2. Simplify \({\log _5}\frac{1}{{125}}\).
Ans:
Given:  \({\log _5}\frac{1}{{125}}\)
\( = {\log _5}1 – {\log _5}125\)
\( = 0 – {\log _5}{5^3}\)
\( = – {\log _5}{5^3}\)
\( = – 3{\log _5}5\)
\({\rm{ = – 3(1)}}\)
\({\rm{ = – 3}}\)
\(\therefore \,{\log _5}\frac{1}{{125}} = – 3\)

Q.3. Solve \({\log _3}(x + 3) = 4\).
Ans: Given: \({\log _3}(x + 3) = 4\)
\( \Rightarrow x + 3 = {3^4}\) [Using, \(y = {\log _a}x\) if and only if \(x = {a^y}\)]
\( \Rightarrow x + 3 = 81\)
\( \Rightarrow x = 78\)
Hence, the required value of \(x\) is \(78\).

Q.4. Write the following expression as single logarithm \(7{\log {12}}x + 2{\log {12}}y\).
Ans: By using the power rule \({\log _b}{M^P} = P{\log _b}M\), we can write the given expression as
\(7{\log _{12}}x + 2{\log _{12}}y = {\log _{12}}{x^7} + {\log _{12}}{y^2}\)
\( = {\log _{12}}\left( {{x^7}{y^2}} \right)\) [By using the product rule \({{{\log }_b}(MN) = {{\log }_b}M + {{\log }_b}N}\)]
Hence, the single logarithm is \({\log _{12}}\left( {{x^7}{y^2}} \right)\)

Q.5. Solve \({\log _x}9 = 2\).
Ans: Given: \({\log _x}9 = 2\)
\( \Rightarrow {x^2} = 9\) [Using, \({y = {{\log }_a}x}\), if and only if \({x = {a^y}}\)]
\( \Rightarrow x = 3\) or \( – 3\)
Since \(x\) is the base, \(x > 0\) and \(x \ne 1\), so \(x = – 3\) is rejected.
Hence, the required solution is \(x = 3\).

Q.6. Write the following expression as single logarithm \(3\log x – 6\log y\).
Ans: By using the power rule \({\log _b}{M^P} = P{\log _b}M\), we can write the given expression as
\(3\log x – 6\log y = \log {x^3} – \log {y^6}\)
\({ = \log \left( {\frac{{{x^3}}}{{{y^6}}}} \right)}\) [By using the quotient rule \({\log _b}\left( {\frac{M}{N}} \right) = {\log _b}M – {\log _b}N\)]
Hence, the single logarithm is \(\log \left( {\frac{{{x^3}}}{{{y^6}}}} \right)\).

Q.7. Write the following expression \(5{\log _9}x + 7{\log _9}y – 3{\log _9}z\) into a single expression
Ans: By using the power rule \({\log _b}{M^P} = P{\log _b}M\), we can write the given expression as
\(5{\log _9}x + 7{\log _9}y – 3{\log _9}z = {\log _9}{x^5} + {\log _9}{y^7} – {\log _9}{z^3}\)
\( = {\log _9}{x^5}{y^7} – {\log _9}{z^3}\) [By using the product rule \({\log _b}(MN) = {\log _b}M + {\log _b}N\)]
 \( = {\log _9}\left( {\frac{{{x^5}{y^7}}}{{{z^3}}}} \right)\) [By using the quotient rule \(\left. {{{\log }_b}\left( {\frac{M}{N}} \right) = {{\log }_b}M – {{\log }_b}N} \right]\)]
Hence, the single logarithm is \(5{\log _9}x + 7{\log _9}y – 3{\log _9}z = {\log _9}\left( {\frac{{{x^5}{y^7}}}{{{z^3}}}} \right)\)

Q.8. Solve \({\log _3}(x + 1) = 2\).
Ans: Given: \({\log _3}(x + 1) = 2\)
\( \Rightarrow x + 1 = {3^2}\)    [Using, \(y = {\log _a}x\) if and only if \(x = {a^y}\)]
\( \Rightarrow x + 1 = 9\)
\( \Rightarrow x = 9 – 1\)
\(\therefore \,x = 8\)
Hence, the required value of \(x\) is \(8\).

Q.9. Solve \({\log _5}(3x – 8) = 2\).
Ans: Given: \({\log _5}(3x – 8) = 2\)
\( \Rightarrow 3x – 8 = {5^2}\) [Using, \(y = {\log _a}x\) if and only if \(x = {a^y}\)]
\( \Rightarrow 3x – 8 = 25\)
\( \Rightarrow 3x = 25 + 8\)
\( \Rightarrow 3x = 33\)
\( \Rightarrow x = \frac{{33}}{3}\)
\(\therefore \,x = 11\)
Hence, the required value of \(x\) is \(11\).

Q.10. Solve \(\log (x + 2) + \log (x – 1) = 1\).
Ans: Given: \(\log (x + 2) + \log (x – 1) = 1\)
\( \Rightarrow \log [(x + 2)(x – 1)] = 1\) [Using, \({\log _b}(MN) = {\log _b}M + {\log _b}N\)]
\( \Rightarrow \log \left( {{x^2} + x – 2} \right) = 1\)
\( \Rightarrow {x^2} + x – 2 = {10^1}\)
\( \Rightarrow {x^2} + x – 12 = 0\)
\( \Rightarrow {x^2} + 4x – 3x – 12 = 0\)
\( \Rightarrow (x + 4)(x – 3) = 0\)
\( \Rightarrow (x + 4) = 0\) and \((x – 3) = 0\)
\( \Rightarrow x = – 4\) and \(x = 3\)
Hence, the required value of \(x\) is \(3\).

Practice Exponentials and Logarithms

Summary

The logarithmic function and exponential function are inverse to each other, i.e. \({\log _b}x = y \rightleftarrows {b^y} = x\). Also learned about the definition of logarithmic function and its graph. Also, the graph of the logarithmic function is obtained with the help of an exponential function. We also observed that the graph of the logarithmic function is either strictly increasing or strictly decreasing, so the domain of the logarithmic function \(x > 0\) or \((0,\,\infty )\) and range is all real numbers \(( – \infty ,\,\infty )\). Later explained the properties of logarithms and solved examples based on logarithms with the help of properties.

Master laws of Logarithm

Frequently Asked Questions (FAQs)

Q.1. What is a logarithmic function? Give an example.
Ans: A function (such as \(y = {\log _a}x\) or \(y = \ln x\)) is inverse of an exponential function (such as \(y = {a^x}\) or \(y = {e^x}\)), so that the independent variable appears in a logarithm. Such functions are called logarithmic functions. For example \({\log _{10}}100 = 2\), since \(100 = {10^2}\).

Q.2. What are the four properties of logarithms?
Ans: (i)  The product rule: \({\log _a}(mn) = {\log _a}(m) + {\log _a}(n)\)
(ii) The quotient rule: \({\log _a}\left( {\frac{m}{n}} \right) = {\log _a}m – {\log _a}n\)
(iii) The power rule: \({\log _a}{M^P} = P{\log _a}M\)
(iv) Change of base formula: \({\log _a}N = \frac{{{{\log }_c}N}}{{{{\log }_c}a}}\)

Q.3. How do you use the properties of logarithms to expand?
Ans: By using the property that \({\log _b}{b^x} = x\). Remember that the properties of exponents and logarithms are very similar. In exponents, to multiply two numbers with the same base, you add the exponents. But in logarithms, the logarithm of a product is the sum of the logarithms.

Q.4. Can you have a negative logarithm?
Ans.:One cannot take the logarithm of a negative number because the domain of logarithmic functions is a set of all non-negative real numbers.

Q.5. Can you square a logarithm?
Ans: For solving a logarithm, logarithm of \({x^n}\) is the same as \(n\) times logarithm of \(x\). So \(n\) can be brought outside the logarithm and multiplied. This can be written as: \({\log _y}{x^n} = n\left( {{{\log }_y}x} \right)\)

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