Exercise 1.5 — Logarithms
Logarithms, their properties and related problems.
Exercise 1.5 — Logarithms: Definition, Properties & Solved Examples
Class 10 Mathematics · CBSE, Telangana & Andhra Pradesh Syllabus · Chapter 1: Real Numbers
Exercise 1.5 is the final exercise of Chapter 1, Real Numbers, in Class 10 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus). It introduces one of the most useful tools in the whole of mathematics — the logarithm — and asks you to apply six fundamental properties of logarithms to evaluate expressions, expand them, rewrite combinations of logs as a single logarithm, and solve equations involving exponents.
Logarithms might look unfamiliar at first, but they are simply a different way of writing the same statement you already know from exponents and powers. Below you'll find the complete definition, every property with its proof, and a fully worked, step-by-step explanation of all the problems in this exercise — ideal for board exam revision.
aˣ = N and the logarithmic form x = logₐ N, every problem in this exercise becomes a matter of applying one of just six properties.
What Is a Logarithm?
If a is a positive number (other than 1) and aˣ = N, then we say that x is the logarithm of N to the base a, and we write this as x = loga N. The two forms — exponential and logarithmic — describe exactly the same relationship between three numbers; only the "subject" of the statement changes.
Every exponential statement aˣ = N can be rewritten instantly as a logarithmic statement x = logₐN, and vice versa.
If aˣ = N, then x = logₐ N — where a > 0, a ≠ 1, N > 0, and a, N are real numbers.
For example, since 2³ = 8, we can immediately say log₂ 8 = 3. Since 10² = 100, we know log₁₀ 100 = 2. Getting comfortable with this back-and-forth conversion is the single most important skill for this exercise.
Common Logarithms and Natural Logarithms
Logarithms can technically be taken to any valid base, but two particular bases are used so often that they have their own names and special notation:
Common Logarithm
- Base = 10
- Written simply as log N (the base 10 is left out)
- Example: log 7, log 16
- Used in scientific scales like pH and decibels
Natural Logarithm
- Base = e (≈ 2.718)
- Written as logₑ N or ln N
- Example: logₑ 5, logₑ 19
- Used heavily in higher mathematics, calculus, and science
In this exercise, whenever you see "log N" with no base written, you should always read it as log₁₀ N — the common logarithm.
- pH scale (chemistry): pH is defined using a common logarithm of the hydrogen-ion concentration — each whole number change in pH represents a tenfold change in acidity.
- Richter scale (earthquakes): Earthquake magnitude is measured on a logarithmic scale, so a magnitude 6 earthquake releases far more than twice the energy of a magnitude 3 one.
- Decibel scale (sound): Sound intensity is also measured logarithmically, which is why a "small" increase in decibels can represent a large real increase in loudness.
The 6 Fundamental Properties of Logarithms
Every problem in this exercise is solved using one (or a combination) of the following six properties. Each one follows directly from the laws of exponents you already know — here is the full proof of each.
logₐ (mn) = logₐ m + logₐ nlogₐ (m/n) = logₐ m − logₐ nlogₐ (mⁿ) = n · logₐ m- logₐ a = 1 — because a¹ = a, so by definition log of a to its own base is 1.
- logₐ 1 = 0 — because a⁰ = 1 for any valid base a, so the log of 1 (in any base) is always 0.
- alogₐ m = m — if logₐ m = x, then by definition aˣ = m, so substituting back gives alogₐ m = m directly.
Solved Examples — Question 1: Determine the Value
In each part below, the strategy is the same: rewrite the number inside the logarithm as a power of the base (using roots, negative exponents, or factorisation), then apply Property 3 followed by Property 4.
log₂₅25^(1/2) = (1/2)·log₂₅25
log₈₁81^(1/4) = (1/4)·log₈₁81
log₂2⁻⁴ = −4·log₂2
logxx^(1/2) = (1/2)·logxx
log₂2⁹ = 9·log₂2
log₁₀10⁻² = −2·log₁₀10
log3/2(3/2)⁻³ = −3·log3/2(3/2)
This one mixes ordinary exponent rules with logarithm Property 6, so it deserves a closer look.
Solved Examples — Question 2: Write as a Single Logarithm
These problems run Properties 1, 2 and 3 in reverse — combining several logarithms into a single "log N" expression, then evaluating it where possible.
Solved Examples — Question 3: Evaluate in Terms of x and y
Here you're given x = log₂3 and y = log₂5, and asked to express other base-2 logarithms in terms of x and y. The trick is to factorise the number inside the log into 2's, 3's and 5's, then convert the product into a sum using Property 1.
Factor tree for 60: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5 — used in part (iii) below.
log₂15 = log₂3 + log₂5
log₂15 − log₂2 = (x+y) − 1
= 2log₂2 + log₂3 + log₂5
= log₂2 + 3log₂3 + 3log₂5
Solved Examples — Question 4: Expand the Logarithm
"Expanding" means going the other way — breaking one complicated logarithm into a sum or difference of several simpler ones, using Properties 1, 2 and 3 together.
= log 2³ + log 5³
= log 2⁷ − log 5⁴
= (1/2)(3 log x − 2 log y)
Solved Example — Question 5: A Proof Using Logarithm Properties
If x² + y² = 25xy, prove that 2 log (x + y) = 3 log 3 + log x + log y.
Solved Example — Question 6: Finding the Value of x/y + y/x
If log[(x+y)/3] = (1/2)(log x + log y), find the value of x/y + y/x.
Solved Example — Question 7: Logarithmic Equation With Two Unknowns
If 2.3ˣ = 0.23ʸ = 1000, find the value of 1/x − 1/y.
Solved Example — Question 8: Solving an Exponential Equation Using Logs
If 2ˣ⁺¹ = 3¹⁻ˣ, find the value of x.
Solved Examples — Question 9: Rational or Irrational?
Is log 2 rational or irrational? Justify your answer.
Is log 100 rational or irrational? Justify your answer.
Common Mistakes to Avoid
- Confusing logₐ(m + n) with logₐ m + logₐ n: There is no property for the logarithm of a sum — the product rule only applies to multiplication, never to addition.
- Forgetting the base when it isn't written: "log N" with no subscript always means base 10 — don't confuse it with "ln N" (base e).
- Mishandling negative exponents: Remember that 1/aⁿ = a⁻ⁿ before applying Property 3 — skipping this step is the most common error in problems like Q1 (iii) and (vii).
- Dropping the conditions on the base: A logarithm is only defined when the base a > 0, a ≠ 1, and the argument N > 0 — always check these conditions hold in equation-solving problems.
- Sign errors while expanding: When expanding a quotient inside a log, every term in the denominator gets a minus sign — easy to miss with three or more terms, as in Q4 (iv).
Quick Reference — All 6 Properties at a Glance
| # | Property | One-Line Reason |
|---|---|---|
| 1 | logₐ(mn) = logₐm + logₐn | Multiplying numbers adds their exponents |
| 2 | logₐ(m/n) = logₐm − logₐn | Dividing numbers subtracts their exponents |
| 3 | logₐ(mⁿ) = n·logₐm | Raising to a power multiplies the exponent |
| 4 | logₐ a = 1 | a¹ = a, by definition |
| 5 | logₐ 1 = 0 | a⁰ = 1 for any base a |
| 6 | a^(logₐ m) = m | The log and the exponential "undo" each other |
What This Lesson Prepares You For
Logarithms wrap up the Real Numbers chapter by tying together everything you've learned about exponents, powers, and the rational/irrational classification of numbers — revisiting the introduction to real numbers alongside this exercise can help connect the two ideas, especially for the rational/irrational proofs in Question 9.
The algebraic manipulation skills you've practised here — combining, expanding, and solving equations with logs — also build the foundation you'll need later for exponential and logarithmic functions, and they reinforce the factorisation techniques used throughout Polynomials and Quadratic Equations, the next two chapters in the Class 10 syllabus.