Introduction
Logarithms are an essential concept in mathematics, particularly in algebra and calculus. They are the inverse operations of exponentiation and are used to solve equations involving exponents. Logarithms simplify complex calculations and have applications in various fields such as science, engineering, and finance. This study note will cover the fundamental aspects of logarithms, their properties, and their applications, specifically tailored for the JEE Main Mathematics syllabus.
Definition of Logarithm
The logarithm of a number is the exponent to which a base must be raised to produce that number. For a number $a$, the logarithm of $x$ to the base $a$ is denoted as $\log_a x$ and is defined as:
$$ \log_a x = y \quad \text{if and only if} \quad a^y = x $$
Common Logarithms
- Base 10 (Common Logarithm): $\log_{10} x$ is often written as $\log x$.
- Base $e$ (Natural Logarithm): $\log_e x$ is written as $\ln x$, where $e \approx 2.71828$.
The base of a logarithm must be a positive number and cannot be 1.
Properties of Logarithms
Logarithms have several properties that make them useful for simplifying expressions and solving equations:
1. Product Rule
$$ \log_a (xy) = \log_a x + \log_a y $$
ExampleIf $\log_2 8 = 3$ and $\log_2 4 = 2$, then: $$ \log_2 (8 \times 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5 $$
2. Quotient Rule
$$ \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y $$
ExampleIf $\log_2 8 = 3$ and $\log_2 4 = 2$, then: $$ \log_2 \left(\frac{8}{4}\right) = \log_2 8 - \log_2 4 = 3 - 2 = 1 $$
3. Power Rule
$$ \log_a (x^y) = y \log_a x $$
ExampleIf $\log_2 8 = 3$, then: $$ \log_2 (8^2) = 2 \log_2 8 = 2 \times 3 = 6 $$
4. Change of Base Formula
$$ \log_a x = \frac{\log_b x}{\log_b a} $$
ExampleTo find $\log_2 8$ using base 10 logarithms: $$ \log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} \approx \frac{0.9031}{0.3010} \approx 3 $$
Solving Logarithmic Equations
Basic Steps
- Isolate the logarithmic term: Ensure the logarithmic term is on one side of the equation.
- Exponentiate to remove the logarithm: Use the definition of logarithms to rewrite the equation in exponential form.
- Solve the resulting equation: Solve for the variable as you would in any algebraic equation.
Solve $\log_2 (x - 1) = 3$:
- Isolate the logarithmic term: $\log_2 (x - 1) = 3$
- Exponentiate: $2^3 = x - 1$
- Solve: $8 = x - 1 \implies x = 9$
Equations with Multiple Logarithms
When dealing with multiple logarithmic terms, use logarithmic properties to combine or separate them as needed.
ExampleSolve $\log_2 x + \log_2 (x - 3) = 3$:
- Combine logarithms: $\log_2 [x(x - 3)] = 3$
- Exponentiate: $2^3 = x(x - 3) \implies 8 = x^2 - 3x$
- Solve the quadratic: $x^2 - 3x - 8 = 0 \implies (x - 4)(x + 2) = 0$
- Solutions: $x = 4$ or $x = -2$ (discard $x = -2$ as $\log_2 (-2)$ is undefined) Therefore, $x = 4$.
Applications of Logarithms
1. pH Calculation in Chemistry
The pH of a solution is calculated using the formula:
$$ \text{pH} = -\log_{10} [\text{H}^+] $$
ExampleIf the concentration of hydrogen ions in a solution is $1 \times 10^{-4}$ M, then: $$ \text{pH} = -\log_{10} (1 \times 10^{-4}) = 4 $$
2. Richter Scale for Earthquakes
The Richter scale measures the magnitude of an earthquake using logarithms:
$$ M = \log_{10} \left(\frac{A}{A_0}\right) $$
where $A$ is the amplitude of seismic waves and $A_0$ is a reference amplitude.
3. Compound Interest in Finance
The formula for compound interest involves logarithms:
$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$
where $A$ is the amount of money accumulated, $P$ is the principal amount, $r$ is the annual interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the time in years.
Tips and Tricks
Tip- Memorize the properties of logarithms as they are frequently used in simplifying and solving equations.
- Practice changing the base of logarithms to become comfortable with the change of base formula.
- Always check the domain of the logarithmic function; the argument must be positive.
Common Mistakes
Common Mistake- Forgetting to check if the argument of the logarithm is positive.
- Misapplying the properties of logarithms, such as incorrectly combining or separating logarithmic terms.
- Ignoring the base of the logarithm, especially when dealing with natural logarithms ($\ln x$) and common logarithms ($\log x$).
Conclusion
Logarithms are a powerful mathematical tool with wide-ranging applications. Understanding their properties, how to manipulate them, and how to solve logarithmic equations is crucial for success in JEE Main Mathematics. Regular practice and familiarity with common pitfalls will help in mastering this topic.
