How to Solve Logarithmic Equations Step by Step
The concept of logarithms plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding logarithms is essential for students facing board exams, JEE/NEET, and anyone exploring applications in science or finance. This page explains logarithms in simple terms, provides key formulas, rules, and common mistakes, and connects you to related concepts and Vedantu resources for further learning.
What Is Logarithm?
A logarithm is defined as the power to which a number called the base must be raised to get another number. For example, in the equation \( 2^3 = 8 \), the logarithm of 8 with base 2 is 3. Written as \( \log_{2}8 = 3 \). You’ll find this concept applied in areas such as exponential equations, calculus, and computer science.
Key Formula for Logarithms
Here’s the standard formula: \( \log_{b}a = x \) means \( b^x = a \), where \( b \) (the base) is a positive real number not equal to 1, \( a \) is any positive real number, and \( x \) is the logarithm.
Laws and Properties of Logarithms
| Law | Formula | Example |
|---|---|---|
| Product Rule | \( \log_b(MN) = \log_b M + \log_b N \) | \( \log_3(5 \times 7) = \log_3 5 + \log_3 7 \) |
| Quotient Rule | \( \log_b(M/N) = \log_b M - \log_b N \) | \( \log_2(8/4) = \log_2 8 - \log_2 4 \) |
| Power Rule | \( \log_b(M^k) = k\log_b M \) | \( \log_5(25^2) = 2\log_5 25 \) |
| Change of Base | \( \log_b M = \dfrac{\log_a M}{\log_a b} \) | \( \log_2 8 = \dfrac{\log_{10} 8}{\log_{10} 2} \) |
| Log of 1 | \( \log_b 1 = 0 \) | \( \log_{10} 1 = 0 \) |
| Log of Base | \( \log_b b = 1 \) | \( \log_4 4 = 1 \) |
Cross-Disciplinary Usage
Logarithms are not only useful in Maths but also play an important role in Physics, Computer Science, Chemistry, and even finance. They help describe phenomena like earthquake intensity (Richter scale), sound intensity (decibels), radioactive decay, and are used in coding algorithms (like binary search). Students preparing for JEE or NEET will see questions involving logarithms in both the Maths and Physics sections.
Step-by-Step Illustration
Let’s solve:
\( \log_3(x) = 4 \)
1. Recognize that \( \log_3(x) = 4 \) means \( 3^4 = x \ )
2. Calculate \( 3^4 = 81 \ )
3. Final Answer: x = 81
Let’s try a property:
\( \log_2(8) + \log_2(4) \)
1. Use the product rule: \( \log_2(8 \times 4) = \log_2(32) \ )
2. \( 2^5 = 32 \) means \( \log_2(32) = 5 \ )
3. Final Answer: 5
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for solving logarithm equations where the result matches a known power of the base. For example, if you see \( \log_{10} 1000 \), quickly recall that 103 = 1000, so the answer is 3. Practicing powers of 2, 3, and 10 up to 6 or 8 helps you answer these almost instantly in exams.
Example Trick: To evaluate \( \log_2(32) \) at a glance, spot that 32 = 25, so the answer is 5.
Tricks like these are practical in competitive exams like NTSE, Olympiads, and JEE. Vedantu’s live classes include more such shortcuts to build your exam confidence and speed.
Try These Yourself
- Solve: \( \log_5(125) \ ).
- Find: \( \log_{10}(10000) \ ).
- Simplify: \( \log_3(9) + \log_3(3) \ ).
- Write the change of base formula for \( \log_7(49) \ ).
- If \( \log_2(x) = 7 \), what is x?
Frequent Errors and Misunderstandings
- Trying to calculate the logarithm of negative numbers or zero (undefined in real numbers).
- Mixing up the base and the number — remember, in \( \log_b(a) \), b is the base, a is the number.
- Using the properties (like product or quotient rule) without checking that all bases are the same.
- Forgetting that \( \log_b(1) = 0 \), no matter the base (as long as b ≠ 1).
Relation to Other Concepts
The idea of logarithms connects closely with topics such as exponents and exponential functions. Mastering logarithms helps you solve equations where the unknown is in the exponent and builds a strong foundation for understanding growth/decay models, calculus, and higher algebra.
Classroom Tip
A quick way to remember the difference between “log” and “ln” is: “log” usually means log base 10 (common logarithm), and “ln” means log base e (natural logarithm, where e ≈ 2.718). Vedantu’s teachers often use “log10” for base 10 and “ln” for base e during live classes so you won’t get confused in exams.
We explored logarithms—from definition, formula, examples, mistakes, and connections to other topics. Continue practicing with Vedantu to become confident in solving logarithm questions and strengthen your mathematics skills for board exams and beyond.
Logarithm Definition and Types | Log Table | Laws of Exponents | Exponents and Powers
FAQs on Logarithms: Meaning, Laws & Solved Examples
1. What is a logarithm in simple terms?
A logarithm answers the question: "To what power must I raise a base to get a specific number?" For example, in 102 = 100, the logarithm (base 10) of 100 is 2. In general, if bx = y, then logby = x. The base (b) is the number being raised to a power, the exponent (x) is the logarithm, and the result (y) is the number whose logarithm we're finding.
2. What are the basic laws of logarithms?
The fundamental laws of logarithms streamline calculations involving multiplication, division, powers, and roots. These include:
• Product Rule: logb(xy) = logbx + logby
• Quotient Rule: logb(x/y) = logbx - logby
• Power Rule: logb(xp) = p logbx
• Change of Base Formula: logbx = (logkx) / (logkb), where k is any valid base.
3. How do you solve logarithmic equations?
Solving logarithmic equations often involves using the properties of logarithms to simplify the equation and isolate the variable. Techniques include applying the product, quotient, and power rules to combine or separate logarithmic terms, then converting to exponential form (if necessary) to solve for the variable. Always check your solution in the original equation to ensure it's valid (i.e., avoids taking the logarithm of a non-positive number).
4. Where are logarithms used in real life?
Logarithms have extensive real-world applications, appearing in various fields:
• Chemistry: Calculating pH values.
• Physics: Measuring sound intensity (decibels) and earthquake magnitudes (Richter scale).
• Finance: Modeling compound interest and exponential growth.
• Computer Science: Algorithm analysis and complexity.
• Engineering: Signal processing and data analysis.
5. What is the difference between 'log' and 'ln'?
The notation 'log' typically refers to the common logarithm, which has a base of 10 (log10x). 'ln' denotes the natural logarithm, whose base is the mathematical constant e (approximately 2.718) (ln x = logex). While both represent logarithmic functions, they differ in their base and consequently, their numerical values for the same argument.
6. How do I change the base of a logarithm?
The change of base formula allows conversion between any valid bases. To change from base b to base k, use: logbx = (logkx) / (logkb). This is particularly useful when using calculators, which typically only have common (base 10) and natural (base e) logarithm functions. Scientific calculators often have a dedicated change-of-base function.
7. What are common mistakes students make with logarithms?
Common errors include:
• Incorrectly applying logarithm rules (e.g., log(x+y) ≠ log x + log y)
• Forgetting that logbx is only defined for x > 0 and b > 0, b ≠ 1
• Mistakes in algebraic manipulation after applying logarithm properties
• Improper use of the change of base formula
• Confusing common and natural logarithms.
8. Can you explain the relationship between logarithms and exponents?
Logarithms and exponents are inverse operations. If bx = y, then logby = x. This means that logarithms "undo" exponentiation, and vice-versa. Understanding this inverse relationship is crucial for solving logarithmic and exponential equations.
9. How are logarithms used in solving exponential equations?
Logarithms are essential for solving exponential equations. By taking the logarithm of both sides of an exponential equation, you can use logarithm properties to simplify the equation and solve for the unknown exponent. The choice of base for the logarithm depends on the specific problem and often involves choosing a base that simplifies the equation.
10. What is the significance of the natural logarithm (ln)?
The natural logarithm (ln x), with base e, is significant because it arises naturally in many mathematical and scientific contexts. Its derivative is simply 1/x, making it crucial in calculus and differential equations. It's also essential in continuous growth and decay models (e.g., population growth, radioactive decay).
11. Why is the base of a logarithm always positive and not equal to 1?
The base (b) must be positive (b > 0) because negative bases can lead to complex or undefined results when raising them to non-integer powers. The base cannot equal 1 (b ≠ 1) because 1 raised to any power is always 1, resulting in a constant function, not a logarithmic function. This restriction ensures the logarithmic function is well-defined and invertible.
