Tips To Solve CAT Logarithms
- Fundamentals of this concept are useful in solving the questions of the other topics by assuming the unknown values as variables. Make sure to cover other inter-related concepts of CAT syllabus. All the inter-related concepts need to be covered to have a good foundation in concepts.
- Be careful of silly mistakes in this topic, as that is how students generally lose marks here. The number of equations needed to solve the given problem equals the number of variables. A linear equation is an equation which gives a straight line when plotted on a graph.
- If you are confused, enrolling in CAT online coaching will help you a long way.
- Linear equations can be of one variable or two variables, or three variables.
- Let a, b, c and d be constants, and x, y, and z are variables. A general form of a single variable linear equation is ax + b = 0.
- A general form of two-variable linear equation is ax + by = c.
- A general form of three-variable linear equation is ax + by + cz = d.
CAT Logarithms PDF
To help CAT aspirants in their preparation, we have made a comprehensive formula PDF containing all the important linear equations that are essential. This PDF includes all the necessary formulas, techniques, and examples required to solve linear equations efficiently. Click on the link below to download the Linear equations formula PDF.
1. Linear Equations Formulae: Solving Linear Equations
For equations of the form ax + by = c and mx + ny = p, find the LCM of b and n.
Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1.
2. Linear Equations Formulae: Straight Lines
Equations with 2 variables: Consider two equations ax + by = c and mx + ny = p. Each of these equations represents two lines on the x-y coordinate plane. The solution of these equations is the point of intersection.
If : This means that both the equations have the same slope but different intercepts, and hence are parallel to each other. There is no point of intersection and no solution.
If : They have different slopes and hence must intersect at some point, resulting in a unique solution.
If : The two lines have the same slope and intercept. Hence, they are the same lines. As they have infinite points common between them, there are infinitely many solutions possible.
Question 1.

If x and y are positive real numbers such that = 4 and , then x + y equals?
Question 2.

For some positive ral number x, if + = 16/ 3 , then the value of is
Question 3.

For a real number x, if 1/ 2 , , and are in arithmetic progression, then the common difference is
Question 4.

The number of distinct integer values of n satisfying < 0, is
Question 5.

If 5 - + 4 = , then 100x equals
Question 6.

log2 [3 + ] - 2 = 0, then 4x equals
Question 7.

For a real number a, if = 4, then a must lie in the range
Question 8.

If y is a negative number such that , then y equals
Question 9.

If log₄ 5 = (log₄ y)(log₆ √5), then y equals
Question 10.

The value of , for 1 < a ≤ b cannot be equal to
Question 11.

equals .
Question 12.

If = A, = -B and = 1/3, then equals
Question 13.

Let x and y be positive real numbers such that log5(x + y) + log5(x - y) = 3, and log2y - log2x = 1 - log23. Then xy equals
Let x and y be positive real numbers such that log5(x + y) + log5(x - y) = 3, and log2y - log2x = 1 - log23. Then xy equals
Question 14.

The real root of the equation + - 21 = 0 is
Question 15.

Let A be a real number. Then the roots of the equation - 4x - = 0 are real and distinct if and only if
Question 16.

If x is a real number, then is a real number if and only if
If x is a real number, then is a real number if and only if
Question 17.

If x is a positive quantity such that 2x = , then x is equal to
If x is a positive quantity such that 2x = , then x is equal to
Question 18.

If log1281 = p, then 3((4-p)/(4+p)) is equal to
If log1281 = p, then 3((4-p)/(4+p)) is equal to
Question 19.

If log2(5 + log3 a) = 3 and log5(4a + 12 + log2 b) = 3, then a + b is equal to
If log2(5 + log3 a) = 3 and log5(4a + 12 + log2 b) = 3, then a + b is equal to
Question 20.

If p3 = q4 = r5 = s6, then the value of logs(pqr) is equal to
If p3 = q4 = r5 = s6, then the value of logs(pqr) is equal to
Question 21.

The smallest integer n for which 4n > 1719 holds, is closest to
The smallest integer n for which 4n > 1719 holds, is closest to
Question 22.

- + - + - + ?
- + - + - + ?
Question 23.

Suppose, log3x = log12y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log6G is equal to:
Suppose, log3x = log12y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log6G is equal to:
Question 24.

The value of log0.008√5 + log√381 – 7 is equal to:
The value of log0.008√5 + log√381 – 7 is equal to:
Question 25.

If x is a real number such that log35 = log5(2 + x), then which of the following is true?
If x is a real number such that log35 = log5(2 + x), then which of the following is true?
Question 26.

If log (2a × 3b × 5c) is the arithmetic mean of log (22 × 33 × 5), log (26 × 3 × 57), and log (2 × 32 × 54), then a equals
If log (2a × 3b × 5c) is the arithmetic mean of log (22 × 33 × 5), log (26 × 3 × 57), and log (2 × 32 × 54), then a equals
Question 27.

If logyx = a × logzy = b × logxz = ab, then which of the following pairs of values for (a, b) is not possible?
If logyx = a × logzy = b × logxz = ab, then which of the following pairs of values for (a, b) is not possible?
Question 28.

If x ≥ y and y > 1, then the value of the expression
(x/y) + (y/x) can never be
If x ≥ y and y > 1, then the value of the expression
(x/y) + (y/x) can never be
Question 29.

Let u = (log2 x)2 – 6 log2 x + 12 where x is a real number. Then the equation xu = 256, has
Let u = (log2 x)2 – 6 log2 x + 12 where x is a real number. Then the equation xu = 256, has
Question 30.

If log3 2, log3 (2x − 5), log3 (2x − 7/2) are in arithmetic progression, then the value of x is equal to
If log3 2, log3 (2x − 5), log3 (2x − 7/2) are in arithmetic progression, then the value of x is equal to
Question 31.

If 1/3 \log_{3} N = 1 + 5 , then
Question 32.

If log10 x - log10 x^(1/2) = 2 logx 10, then a possible value of x is given by:
If log10 x - log10 x^(1/2) = 2 logx 10, then a possible value of x is given by:
Question 33.

What is the sum of n terms in the series
log m + log (m^2 / n) + log (m^3 / n^2) + log (m^4 / n^3) .......+ log (m^n / n^(n-1)) ?
What is the sum of n terms in the series
log m + log (m^2 / n) + log (m^3 / n^2) + log (m^4 / n^3) .......+ log (m^n / n^(n-1)) ?
Question 34.

Directions: Each question is followed by two statements I and II. Mark:
1. if the question can be answered by any one of the statements alone, but cannot be answered by using the other statement alone.
2. if the question can be answered by using either statement alone.
3. if the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
4. if the question cannot be answered even by using both the statements together.
What is the distance x between two cities A and B in integral number of kilometres?
I. x satisfies the equation =
II. x ≤ 10 km
Directions: Each question is followed by two statements I and II. Mark:
1. if the question can be answered by any one of the statements alone, but cannot be answered by using the other statement alone.
2. if the question can be answered by using either statement alone.
3. if the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
4. if the question cannot be answered even by using both the statements together.
What is the distance x between two cities A and B in integral number of kilometres?
I. x satisfies the equation =
II. x ≤ 10 km
Question 35.

If log2 [log7 (x2 - x + 37)] = 1, then what could be the value of ‘x’?
If log2 [log7 (x2 - x + 37)] = 1, then what could be the value of ‘x’?
Question 36.

If log7 log5 ((x+5)^(1/2) + x^(1/2)) = 0, find the value of x.
If log7 log5 ((x+5)^(1/2) + x^(1/2)) = 0, find the value of x.
Question 37.

log6 216 is
log6 216 is