Exponential Functions
- Characteristics of Exponential Functions
- Examples of Exponential Functions in Real Life
Logarithmic Functions
- Characteristics of Logarithmic Functions
- Examples of Logarithmic Functions in Real Life
The Connection Between Exponential and Logarithmic Functions
Properties of Logarithms
Graphs of Exponential and Logarithmic Functions
- Exponential Function Graph
- Logarithmic Function Graph
Applications of Exponential and Logarithmic Functions
- Exponential Functions
- Logarithmic Functions
Solving Problems with Exponentials and Logarithms
- Example 1: Solving an Exponential Equation
- Example 2: Solving a Logarithmic Equation
FAQs on Logarithmic and Exponential functions

Logarithmic and Exponential functions

Exponential Functions

An exponential function is a mathematical expression where a constant base is raised to the power of a variable. It’s written in the form:

f(x) = a \cdot b^x

$f$ $($ $x$ $)$ $=$ $a$ $\cdot$ $b$ $x$

Fill out the form for expert academic guidance

$a$ $a$ : This is the starting value or initial amount.
$b$ $b$ : The base, a positive number that determines the rate of change.
$x$ $x$ : The variable or the exponent.

Characteristics of Exponential Functions

Exponential Growth:
- If $b > 1$ $b$ $>$ $1$ , the function grows rapidly as $x$ $x$ increases.
- Example: Population growth, where each year’s population is larger than the last.
- Formula for population growth: $P(t) = P_0 \cdot (1 + r)^t$ $P$ $($ $t$ $)$ $=$ $P$ $0$ $$ $\cdot$ $($ $1$ $+$ $r$ $)$ $t$ , where $P_0$ $P$ $0$ $$ is the initial population, $r$ $r$ is the growth rate, and $t$ $t$ is time.
Exponential Decay:
- If $0 < b < 1$ $0$ $<$ $b$ $<$ $1$ , the function decreases as $x$ $x$ increases.
- Example: Radioactive decay, where the substance reduces over time.
- Formula for decay: $N(t) = N_0 \cdot e^{-kt}$ $N$ $($ $t$ $)$ $=$ $N$ $0$ $$ $\cdot$ $e$ $-$ $kt$ , where $N_0$ $N$ $0$ $$ is the initial amount, $k$ $k$ is the decay constant, and $t$ $t$ is time.
Asymptote:
- The graph of an exponential function approaches the x-axis but never touches it.

Examples of Exponential Functions in Real Life

Compound Interest: If you invest money in a bank account that compounds interest, the amount grows exponentially. For example, if you invest $1,000 at 5% interest per year, the formula is:

$A(t) = 1000 \cdot (1.05)^t$ $A$ $($ $t$ $)$ $=$ $1000$ $\cdot$ $($ $1.05$ $)$ $t$

Unlock the full solution & master the concept

Get a detailed solution and exclusive access to our masterclass to ensure you never miss a concept

Population Growth: If a population doubles every year, you can model it as:

$P(t) = P_0 \cdot 2^t$ $P$ $($ $t$ $)$ $=$ $P$ $0$ $$ $\cdot$ $2$ $t$

Ready to Test Your Skills?

Check Your Performance Today with our Free Mock Tests used by Toppers!

Take Free Test

Disease Spread: During an outbreak, the number of infected individuals grows exponentially initially, making this function crucial in predicting the spread.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It helps us find the power to which a base must be raised to get a certain number. The general form is:

f(x) = \log_b(x)

$f$ $($ $x$ $)$ $=$ $log$ $b$ $$ $($ $x$ $)$

🔥 Start Your JEE/NEET Prep at Just ₹1999 / month - Limited Offer! Check Now!

$b$ $b$ : The base, a positive number $b > 0, b \neq 1$ $b$ $>$ $0$ $,$ $b$ $\neq$ $=$ $1$ .
$x$ $x$ : The input, which must be positive ( $x > 0$ $x$ $>$ $0$ ).
$\log_b(x)$ $log$ $b$ $$ $($ $x$ $)$ : The output, which is the exponent.

Characteristics of Logarithmic Functions

Inverse Relationship:
- If $b^y = x$ $b$ $y$ $=$ $x$ , then $\log_b(x) = y$ $log$ $b$ $$ $($ $x$ $)$ $=$ $y$ .
- Example: If $2^3 = 8$ $2$ $3$ $=$ $8$ , then $\log_2(8) = 3$ $log$ $2$ $$ $($ $8$ $)$ $=$ $3$ .
Domain and Range:
- The domain of logarithmic functions is $x > 0$ $x$ $>$ $0$ (logarithms of zero or negative numbers are undefined).
- The range is all real numbers.
Graph Features:
- The graph has a vertical asymptote at $x = 0$ $x$ $=$ $0$ .
- It passes through the point $(1, 0)$ $($ $1$ $,$ $0$ $)$ because $\log_b(1) = 0$ $log$ $b$ $$ $($ $1$ $)$ $=$ $0$ .

Examples of Logarithmic Functions in Real Life

Earthquakes: The Richter scale measures the magnitude of earthquakes logarithmically. For example, a magnitude 6 earthquake is 10 times stronger than a magnitude 5.
Sound: Decibels (dB) measure sound intensity. A 10 dB increase means the sound is 10 times more intense.
pH Scale: In chemistry, the acidity or alkalinity of a solution is measured logarithmically. A solution with pH 4 is 10 times more acidic than one with pH 5.

The Connection Between Exponential and Logarithmic Functions

Exponential and logarithmic functions are two sides of the same coin. They are inverses, meaning one undoes the other. This is expressed as:

$b^{\log_b(x)} = x$ $b$ $log$ $b$ $$ $($ $x$ $)$ $=$ $x$
$\log_b(b^x) = x$ $log$ $b$ $$ $($ $b$ $x$ $)$ $=$ $x$

Example:

create your own test

YOUR TOPIC, YOUR DIFFICULTY, YOUR PACE

start learning for free

If $b = 2$ b=2:\n
- $2^{\log_2(8)} = 8$ $2$ $log$ $2$ $$ $($ $8$ $)$ $=$ $8$
- $\log_2(2^4) = 4$ $log$ $2$ $$ $($ $2$ $4$ $)$ $=$ $4$

Properties of Logarithms

Logarithms follow specific rules, making them easier to work with:

Product Rule:

$\log_b(x \cdot y) = \log_b(x) + \log_b(y)$ $log$ $b$ $$ $($ $x$ $\cdot$ $y$ $)$ $=$ $log$ $b$ $$ $($ $x$ $)$ $+$ $log$ $b$ $$ $($ $y$ $)$

Quotient Rule:

$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$ $log$ $b$ $$ $($ $yx$ $$ $)$ $=$ $log$ $b$ $$ $($ $x$ $)$ $-$ $log$ $b$ $$ $($ $y$ $)$

Power Rule:

$\log_b(x^k) = k \cdot \log_b(x)$ $log$ $b$ $$ $($ $x$ $k$ $)$ $=$ $k$ $\cdot$ $log$ $b$ $$ $($ $x$ $)$

Change of Base Formula:

$\log_b(x) = \frac{\log(x)}{\log(b)}$ $log$ $b$ $$ $($ $x$ $)$ $=$ $log$ $($ $b$ $)$ $log$ $($ $x$ $)$ $$

Graphs of Exponential and Logarithmic Functions

Exponential Function Graph

Starts slow for negative $x$ $x$ , then grows rapidly for positive $x$ $x$ .
Has a horizontal asymptote at $y = 0$ $y$ $=$ $0$ .

Logarithmic Function Graph

Grows rapidly for small $x$ $x$ , then slows down as $x$ $x$ increases.
Has a vertical asymptote at $x = 0$ $x$ $=$ $0$ .

When plotted together, their graphs are reflections of each other across the line $y = x$ $y$ $=$ $x$ .

Applications of Exponential and Logarithmic Functions

Exponential Functions

Finance: Predicting investment growth.
Biology: Modeling population or bacterial growth.
Physics: Describing radioactive decay.

Logarithmic Functions

Earth Sciences: Measuring earthquakes.
Sound Engineering: Determining sound levels.
Medicine: Understanding pH levels.

Solving Problems with Exponentials and Logarithms

Example 1: Solving an Exponential Equation

Solve $3^x = 81$ $3$ $x$ $=$ $81$ :

Rewrite $81$ $81$ as a power of 3:

$3^x = 3^4$ $3$ $x$ $=$ $3$ $4$

Equate exponents:

$x = 4$ $x$ $=$ $4$

Example 2: Solving a Logarithmic Equation

Solve $\log_2(x) = 5$ $log$ $2$ $$ $($ $x$ $)$ $=$ $5$ :

Rewrite in exponential form:

$x = 2^5$ $x$ $=$ $2$ $5$

Simplify:

$x = 32$ $x$ $=$ $32$

FAQs on Logarithmic and Exponential functions

What is the difference between exponential and logarithmic functions?

The exponential function can be represented by ƒ(x) = ex, while the logarithmic function is represented by g(x) = ln x, with the former being the inverse of the latter. The exponential function's domain is indeed a set of real numbers, whereas the logarithmic function's domain is a set of positive real numbers.

What is exponential and logarithmic functions used for?

Interest earned on an investment, population growth, and carbon dating seems to be three of the most common applications of exponential and logarithmic functions.

How do you solve logarithmic and exponential functions?

For solving an exponential equation, first isolate the exponential expression, then logarithm on both sides of the equation, and solve for the variable. For solving a logarithmic equation, isolate the logarithmic expression first, then exponentiate both sides of the equation and solve for the variable.