CAT Logarithms & Indices questions (2023–2025) — actual Quant questions with solutions

The sum of all possible real values of x for which log_x-3(x² − 9) = log_x-3(x + 1) + 2 is :

X Maro G Strategy | Surds and Indices | Moderate

If 12¹²^x × 4²⁴^{x +}¹² × 5²^y = 8⁴^z × 20¹²^x × 243^3x-6 , where x, y and z are natural numbers, then x+y+z equals_____

The sum of digits of the number (625)⁶⁵ × (128)³⁶ is ______

If log₆₄(x²) + log₈(√y) + 3 log₅₁₂(√(y)z) = 4, where x, y and z are positive real numbers, then the minimum possible value of (x + y + z) is

Concepts of Indices | HARD | CAT 2025 Slot 1

For any natural number k, let aₖ = 3ᵏ. The smallest natural number m for which (a₁)¹ × (a₂)² × ... × (a₂₀)²⁰ < (a₂₁ × a₂₂ × ... × a₂₀₊ₘ)⁵ is:

The number of distinct integers n for which log_1/4 of (n² − 7n + 11) > 0 is _____.

Indices + Logs

If 3^a = 4, 4^b = 5, 5^c = 6, 6^d = 7, 7^e = 8, and 8^f = 9, then what is the value of abcdef?

\( a \) and \( b \) are natural numbers, then the value of \( a + b \) is: \[ (a + b\sqrt{3})^2 = 52 + 30\sqrt{3} \]

Logs + Maxima

Logarithm Expression Problem If \( a, b, c \) are positive real numbers such that: \[ a > 10 \geq b \geq c \quad \text{and} \quad \frac{\log_8(a + b)}{\log_2 c} + \frac{\log_{27}(a - b)}{\log_3 c} = \frac{2}{3} \] Then the greatest possible integer value of \( a \) is?

Logs DPAC Shortcut

Logarithm Equation Question If \( x \) is a positive real number such that \( 4 \log_{10} x + 4 \log_{100} x + 8 \log_{1000} x = 13 \), then the greatest integer not exceeding \( x \) is:

If a > 10 ≥ b ≥ c and given a logarithmic equation involving a, b, c, find the greatest possible integer value of a.

If 3^a = 4, 4^b = 5, 5^c = 6, 6^d = 7, 7^e = 8, 8^f = 9, then find the value of a×b×c×d×e×f.

Solve: 10^x + 4/(10^x) = 81/2. Find sum of all real values of x.

The sum of all real values of k for which the equation x²²⁷⁶ = x²²⁷⁶ holds true is. If k = 1, 32768, then the value of k is

If x is a positive real number such that 4 log₁₀ x + 4 log₁₀₀ x + 8 log₁₀₀₀ x = 13, then the greatest integer not exceeding x is

Let \( n \) be any natural number such that \( 5^{n-1} < 3^{n+1} \). The least integer value of \( m \) that satisfies \( 3^{n+1} < 2^{n+m} \) for all such \( n \) is?