Exponential and Logarithmic Functions:  Exponential functions are primarily employed to calculate population growth, compound interest, and radioactivity. Real-world applications such as bacterial growth/decay, population growth/decline, and compound interest are frequently represented as exponential functions.

Economists, engineers, computer programmers, and many others use exponential functions. Logarithmic functions are inverse of exponential functions and have a range of applications. Logarithmic scales measure quantities with broad values because small exponents can correspond to very large powers. Richter scale, for example, measures the logarithm of a quake’s intensity, whereas loudness is measured in Decibels, which are the logarithm of the power transmitted by a sound wave.

Exponential Function

The exponent of a number shows how many times a number is multiplied. Any number can be multiplied by itself as many times as you want, and this is expressed using exponents. If the exponent is \(1\), then it is the number itself. If the exponent is \(0\), then the value is \(1\).

The term ‘exponent’ means the ‘power’ of a number. The exponent of \(3\) in the number \(3^{4}\) is equal to \(4\). The exponential functions are those with the variable occurring as a power. An exponential function can be defined as:

\(f(x)=a^{x}\)

where \(a\) is the base and is a positive real number, not equal to \(1\).

If \(a=1\), then \(f(x)=1^{x}\), which is equal to \(1, \forall x\).

The most used exponential function base is the transcendental number \(e\), and the value of \(e\) is equal to \(2.71828\).

Using the base as “\(e\)” we can represent the exponential function as

\(y=e^{x}\)

This is referred to as the natural exponential function. The common exponential function, on the other hand, is an exponential function with base \(10\).

Graph of Exponential Functions

Depending on the value of ‘\(a\)’, we can have two possible cases:

Case 1: \(a>1\)

Here, the exponential function increases very rapidly with increasing \(x\) and tends to \( + \infty \) as \(x\) tends to \( + \infty \).

• When \(x=0, a^{x}=1\)
• When \(x\) tends to \(-\infty\), the function tends to \(0\).

The general graph of the function is as shown below. (where \(a=2\))

Case 2: \(a<1\)

The function decreases very rapidly with increasing \(x\) and tends to \(0\) as \(x\) tends to \( + \infty \).

• When \(x=0, a^{x}=1\)
• When \(x\) tends to \(-\infty\), the function tends to \(+\infty\)

The general graph of such a function looks as shown below. (where \(a=2\))

Properties of Exponential Functions

• The domain of the exponential function is \((-\infty,+\infty)\) i.e., it is defined for all \(x(\forall x)\).
• The range of the exponential function is \((0,+\infty)\).

The function value is never negative; it is not even \(0\). This property should be clear from the graph of the function \({a^x}\). Also, it is logical that the power of any real number can’t be a negative number.

• The points \((0,1)\) and \((1, a)\) always lie on the graph of \(a^{x}\).
• \(‘ a^{\prime}\) must necessarily be a positive number. If \(a\) is a negative number, then for any fractional values of \(x\), we will get an imaginary number as a result which can’t be plotted on the same graph. Example: \((-2)^{\frac{1}{2}}=\sqrt{2 i}\)
• Product rule: \(a^{x} \cdot a^{y}=a^{x+y}\)
• Quotient rule: \(\frac{a^{x}}{a^{y}}=a^{x-y}\)
• The standard exponential function \(e^{x}\) is a unique function as it is equal to its derivative.

\(\frac{d}{d x} e^{x}=e^{x}\)

Logarithmic Functions

The logarithm and exponential functions are inverse functions of each other. The logarithm function does the opposite of taking the power of a number.

Exponential Form : \({a^y} = x\)

Logarithmic Form: \(y=\log _{a} x\)

With these two forms, you can easily see that the value of the function \(f(x)=\log _{a} x\) is the power to which \(a\) must be raised to get \(x\). Therefore, \(x\) cannot be negative as that would require \(a\) to be imaginary.

The conditions on the base \(a\) are:

• \(a>0\) : It follows directly from the exponential representation of the logarithmic function.
• \(a \neq 1\) : As \(1\) raised to any power would only give \(1\).

Hence, there are two possible cases:

Case 1: \(a>1\)

• The logarithmic function decreases rapidly with decreasing \(x\) and tends to \(-\infty\), as \(x\) tends to \(0\).
• When \(x\) tends to \(+\infty\), the function also tends to \(+\infty\) with an ever-decreasing rate of increase.

The general graph of the function looks as shown below. (where \(a=2\))

Case 2: \(0<b<1\)

• The function increases rapidly to \(+\infty\), as \(x\) tends to \(0\)
• Falls at an ever-decreasing rate to \(-\infty\), as \(x\) tends to \(+\infty\).

The general graph is shown below. (where \(a=0.5\))

Properties of Logarithmic Functions

• Domain of the logarithmic functions is \((0,+\infty)\).
• Range of the logarithmic function is \((-\infty,+\infty)\).
• The points \((1,0)\) and \((a, 1)\) always lie on the graph of the function \(\log _{a} x\).

• Product rule: \(\log _{a} x y=\log _{a} x+\log _{a} y\)
• Quotient rule: \(\log _{a} \frac{x}{y}=\log _{a} x-\log _{a} y\)
  Power rule: \(\log _{a} b^{x}=x \log _{a} b\)
  Change of base formula: To change the logarithm from a given base ‘\(a\)’ to base ‘\(b\)’ \(\log _{a} x=\frac{\log _{b} x}{\log _{b} a}\)
• In \(e=1\)

Relation Between Exponential and Logarithmic Functions

The logarithmic and exponential functions are inverses of one another. This can be confirmed by looking at the properties.

• The two functions’ ranges and domains are swapped.

• The points \((0,1)\) and \((1,a)\) are always on the graph of the exponential function, whereas \((1,0)\) and \((a,1)\) are always on the graph of the logarithmic function.
• The product and quotient rules of exponential and logarithmic functions are interconnected.

Solved Examples on Exponential and Logarithmic Functions

Q1. Express the given exponential function \(8^{2}=64\) to its equivalent logarithmic function.
Ans: Given exponential function is \(8^{2}=64\).
Here,
Base \(=8\)
Exponent \(=2\)
Argument \(=64\)
Therefore, \(8^{2}=64\) in logarithmic function is \(\log _{8} 64=2\)

Q2. Solve the exponential function \(\frac{1}{5^{2 x-4}}=125\).
Ans: Given: \(\frac{1}{5^{2 x-4}}=125\)
\(\Rightarrow 5^{-2 x+4}=5^{3}\)
\(\Rightarrow-2 x+4=3\)
\(\Rightarrow-2 x=-1\)
\(\Rightarrow 2 x=1\)
\(\therefore x=\frac{1}{2}\)

Q3. Solve \(2^{x}+3^{x-2}=3^{x}-2^{x+1}\)
Ans: Given: \(2^{x}+3^{x-2}=3^{x}-2^{x+1}\)
\(\Rightarrow 2^{x}+2^{x+1}=3^{x}-3^{x-2}\)
\(\Rightarrow 2^{x}+2^{x} \times 2=3^{x}-3^{x} \times 3^{-2}\)
\(\Rightarrow 2^{x}(1+2)=3^{x}\left(1-\frac{1}{9}\right)\)
\(\Rightarrow 2^{x} \times 3=3^{x}\left(\frac{8}{9}\right)\)
\(\Rightarrow 2^{x}=3^{x}\left(\frac{8}{3 \times 9}\right)\)
\(\Rightarrow 2^{x}=3^{x}\left(\frac{8}{27}\right)\)
\(\Rightarrow \frac{2^{x}}{3^{x}}=\frac{8}{27}\)
\(\Rightarrow \frac{2^{x}}{3^{x}}=\frac{2^{3}}{3^{3}}\)
\(\therefore x=3\)

Q4. Plot the graphs for

(i) \(f\left( x \right) = {10^x}\)	(ii) \(f\left( x \right) = {3^x}\)	(iii) \(f\left( x \right) = {2^x}\)	(iv) \(f\left( x \right) = {1.5^x}\)
(v) \(f\left( x \right) = {1.25^x}\)	(vi) \(f\left( x \right) = {1^x}\)	(vii) \(f\left( x \right) = {\frac{1}{2}^x}\)

What inference you can make from these graphs?

Ans: The graphs can be plotted as given below.

As we know, \(1\) raised to any power will give the result as \(1\) and hence \(f\left( x \right) = {1^x}\) will have the value equal to \(1\) always and hence the graph will be a straight line parallel to \(x\)-axis, \(y=1\).

In \(f(x)=\frac{1}{2}^{x}\), we can see that the value of \(a\) is less than\(1\) and hence the function will be decreasing in nature.

All other functions will be increasing. The function will grow faster as the value of \(a\) increases.

Q5. If \(2 \log x=4 \log 3\), then find the value of ‘\(x\)’.
Ans: Given: \(2 \log x=4 \log 3\)
Divide each side by \(2\)
\(\frac{2 \log x}{2}=\frac{4 \log 3}{2}\)
\(\Rightarrow \log x=2 \log 3\)
\(\Rightarrow \log x=\log 3^{2}\)
\(\Rightarrow \log x=\log 9\)
\(\therefore x=9\)

Summary

An exponential function is of the form \(f(x)=b^{y}\), where \(b>0\) and \(b \neq 1\). The quantity \(x\) is the number, \(b\) is the base, and \(y\) is the exponent or power. The logarithmic function is given by \(f(x)=\log _{b} x=y\), where \(b\) is the base, \(y\) is the exponent, and \(x\) is the argument. The function \(f(x)=\log _{b} x\) is read as, “log of \(x\) to the base \(b\) “. The natural \(\log\) or \(\ln\) is the inverse of the exponential function \(e\), that is, \(\ln e=1\). Complex exponential or logarithmic functions can be simplified using the exponential or logarithmic properties.

FAQs on Exponential and Logarithmic Functions

Q1. How do you tell if a function is exponential or logarithmic?
Ans: If the function is of the form \(f(x)=b^{y}\), where \(b>0\) and \(b \neq 1\), then it is an exponential function. The quantity \(x\) is the number, \(b\) is the base, and \(y\) is the exponent or power. If the function is of the form \(f(x)=\log _{b} x=y\), then this is a logarithmic function, where \(b\) is the base, \(y\) is the exponent, and \(x\) is the argument.

Q2. How do you simplify exponential and logarithmic functions?
Ans: Complex exponential or logarithmic functions can be simplified using the exponential or logarithmic laws.

Q3. What is an exponential function and example?
Ans: An exponential function is of the form \(f(x)=b^{y}\), where \(b>0\) and \(b \neq 1\). The quantity \(x\) is the number, \(b\) is the base, and \(y\) is the exponent or power. Example: \(2^{x}, 5^{x}, 21^{x}\), etc.

Q4. What is an exponential function in your own words?
Ans: An exponential function is function of the form \({a^x}\), where \(a\) is the base and \(a>0\) and \(a \ne 0\).

Q5. What is the difference between exponential and logarithmic functions?
Ans: The exponential function is given by \(f(x)=a^{x}\), where \(a>0\) and \(a \neq 1\). Whereas the logarithmic function is given by \(g(x)=\ln x\). The exponential functions and logarithmic functions are inverse to each other.

Learn About the Law of Exponents Here

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