Basics & Logarithms81 questions23 PYQ

Basics & Logarithms — JEE Maths Practice Questions & Solutions

81 questions on Basics & Logarithms with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

mediumPYQ · JEE Main 2024
The sum of all the solutions of the equation (8)2x−16⋅(8)x+48=0(8)^{2x}-16\cdot(8)^{x}+48=0 is
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mediumPYQ · JEE Main 2024
The number of distinct real roots of the equation ∣x+1∣∣x+3∣−4∣x+2∣+5=0|x+1||x+3|-4|x+2|+5=0 is
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mediumPYQ · JEE Main 2021
The number of elements in the set {x∈R:(∣x∣−3)∣x+4∣=6}\{x\in\mathbb{R}:(|x|-3)|x+4|=6\} is equal to
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mediumPYQ · JEE Main 2023
The number of elements in the set {n∈Z:∣n2−10n+19∣<6}\{n\in\mathbb{Z}:|n^{2}-10n+19|<6\} is
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mediumPYQ · JEE Main 2021
The value of 3+14+13+14+13+…∞3+\dfrac{1}{4+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{3+\ldots\infty}}}} is equal to
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mediumPYQ · JEE Main 2021
The number of solutions of the equation log⁡4(x−1)=log⁡2(x−3)\log_{4}(x-1)=\log_{2}(x-3) is
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mediumPYQ · JEE Main 2025
The sum of the squares of the roots of ∣x−2∣2+∣x−2∣−2=0|x-2|^{2}+|x-2|-2=0 and the squares of the roots of x2−2∣x−3∣−5=0x^{2}-2|x-3|-5=0 is
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mediumPYQ · JEE Main 2024
The number of real solutions of the equation x∣x+5∣+2∣x+7∣−2=0x|x+5|+2|x+7|-2=0 is
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mediumPYQ · JEE Main 2023
The number of integral solutions xx of log⁡(x+7/2)(x−72x−3)2≥0\log_{(x+7/2)}\left(\dfrac{x-7}{2x-3}\right)^{2}\ge 0 is
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mediumPYQ · JEE Main 2022
The sum of all the real roots of the equation (e2x−4)(6e2x−5ex+1)=0(e^{2x}-4)(6e^{2x}-5e^{x}+1)=0 is
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mediumPYQ · JEE Main 2021
The number of real roots of the equation (x+1)2+∣x−5∣=274(x+1)^{2}+|x-5|=\dfrac{27}{4} is
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mediumPYQ · JEE Main 2021
The sum of the roots of the equation x+1−2log⁡2(3+2x)+2log⁡4(10−2−x)=0x+1-2\log_{2}(3+2^{x})+2\log_{4}(10-2^{-x})=0 is
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mediumPYQ · JEE Main 2021
The number of solutions of the equation log⁡(x+1)(2x2+7x+5)+log⁡(2x+5)(x+1)2−4=0\log_{(x+1)}(2x^{2}+7x+5)+\log_{(2x+5)}(x+1)^{2}-4=0, x>0x>0 is
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mediumPYQ · JEE Main 2020
Let SS be the set of all real roots of the equation 3x(3x−1)+2=∣3x−1∣+∣3x−2∣3^{x}(3^{x}-1)+2=|3^{x}-1|+|3^{x}-2|. Then SS
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mediumPYQ · JEE Main 2020
The value of (0.16)log⁡2.5 ⁣(13+132+133+⋯ to ∞)(0.16)^{\log_{2.5}\!\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\cdots\text{ to }\infty\right)} is equal to
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mediumPYQ · JEE Main 2019
The number of real roots of the equation 5+∣2x−1∣=2x(2x−2)5+|2^{x}-1|=2^{x}(2^{x}-2) is
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mediumPYQ · JEE Main 2016
The sum of all real values of xx satisfying the equation (x2−5x+5)x2+4x−60=1(x^{2}-5x+5)^{x^{2}+4x-60}=1 is
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mediumPYQ · JEE Main 2025
The number of real roots of the equation x∣x−2∣+3∣x−3∣+1=0x|x-2|+3|x-3|+1=0 is
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mediumPYQ · JEE Main 2026
If the set of all solutions of ∣x2+x−9∣=∣x∣+∣x2−9∣|x^2+x-9|=|x|+|x^2-9| is [α,β]∪[γ,∞)[\alpha,\beta]\cup[\gamma,\infty), then (α2+β2+γ2)(\alpha^2+\beta^2+\gamma^2) is equal to
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mediumPYQ · JEE Advanced 2011
Let (x0,y0)(x_{0},y_{0}) be the solution of the following equations: (2x)ln⁡2=(3y)ln⁡3(2x)^{\ln 2}=(3y)^{\ln 3} and 3ln⁡x=2ln⁡y3^{\ln x}=2^{\ln y}. Then x0x_{0} is
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easyPYQ · JEE Main 2025
The product of all solutions of the equation e5(log⁡ex)2+3=x8e^{5(\log_{e}x)^{2}+3}=x^{8}, x>0x>0, is
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easyPYQ · JEE Main 2024
The number of real solutions of the equation x(x2+3∣x∣+5∣x−1∣+6∣x−2∣)=0x\left(x^{2}+3|x|+5|x-1|+6|x-2|\right)=0 is
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easyPYQ · JEE Main 2021
If a+b+c=1a+b+c=1, ab+bc+ca=2ab+bc+ca=2 and abc=3abc=3, then the value of a4+b4+c4a^{4}+b^{4}+c^{4} is equal to
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hard
Let x=3−5x = \sqrt{3-\sqrt{5}} and y=3+5y = \sqrt{3+\sqrt{5}}. If the value of the expression x−y+2x2y+2xy2−x4y+xy4x - y + 2x^2y + 2xy^2 - x^4y + xy^4 can be expressed in the form p+q\sqrt{p}+\sqrt{q} (where p,q∈Np, q \in \mathbb{N}), then the value of (p+q)(p+q) is:
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hard
If ax=ba^{x} = b, by=cb^{y} = c, cz=ac^{z} = a (where a,b,c,x,y,za, b, c, x, y, z are positive real numbers), then the value of
∑r=189log⁡10tan⁡(rπ180)xyz\frac{\displaystyle\sum_{r=1}^{89}\log_{10}\tan\left(\frac{r\pi}{180}\right)}{xyz}
 is equal to
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hard
If x=4−23x = \sqrt{4-2\sqrt{3}} and y=9−45y = \sqrt{9-4\sqrt{5}}, then the value of (5 x−3 y)2(\sqrt{5}\,x - \sqrt{3}\,y)^2 is equal to (a−bc)(a - b\sqrt{c}), where a,b,ca, b, c are co-prime and cc is an odd integer. Then a+b+ca+b+c is equal to:
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hard
If x,y,z∈Rx, y, z \in \mathbb{R} and 121x2+4y2+9z2−22x+4y+6z+3=0121x^2 + 4y^2 + 9z^2 - 22x + 4y + 6z + 3 = 0, then the value of 1x−1y−1z\dfrac{1}{x}-\dfrac{1}{y}-\dfrac{1}{z} is equal to:
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hard
If λ\lambda is the minimum value of ∣x−p∣+∣x−15∣+∣x−p−15∣|x-p| + |x-15| + |x-p-15| for xx in the range p≤x≤15p \leq x \leq 15 where 0<p<150 < p < 15, then λ5\dfrac{\lambda}{5} is
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hard
Find the number of single digit positive integers satisfying ∣3xx2−4∣≥1\left|\dfrac{3x}{x^2-4}\right| \geq 1 is
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hard
If A=(log⁡31−log⁡34)(log⁡39−log⁡32)(log⁡31−log⁡39)(log⁡38−log⁡34)A = \dfrac{(\log_3 1 - \log_3 4)(\log_3 9 - \log_3 2)}{(\log_3 1 - \log_3 9)(\log_3 8 - \log_3 4)}, then the value of 2(3A)2(3^A) is
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hard
Find the modulus of the sum of all solutions to (x2+3x+1)x2+2x−8=1(x^2+3x+1)^{x^2+2x-8} = 1 is
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hard
If log⁡α8=γ\log_{\alpha}8 = \gamma, log⁡βα=−1\log_{\beta}\alpha = -1 and log⁡1/4β=−1\log_{1/4}\beta = -1 then
(1α+1)log⁡5(β2+4γ2)\left(\frac{1}{\alpha}+1\right)^{\log_{\sqrt5}\left(\beta^{2}+4\gamma^{2}\right)}
 is equal to
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hard
If (1+1x)x+1=(1+12021)2021\left(1+\dfrac1x\right)^{x+1} = \left(1+\dfrac{1}{2021}\right)^{2021}, then ∣x∣\left|x\right| is equal to
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hard
The greatest value of ln⁡(x)ln⁡(z)\ln\left(x\right)\ln\left(z\right) provided that ln⁡(x)+log⁡y(z)=3\ln\left(x\right)+\log_y\left(z\right) = 3 and ln⁡(y)+log⁡x(z)=4\ln\left(y\right)+\log_x\left(z\right) = 4 where x,y,z>1x,y,z>1.
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hard
Number of solution(s) of the equation log⁡2(x2+3)=12log⁡1/3(x+1x)\log_{2}\left(x^{2}+3\right) = \dfrac12\log_{1/3}\left(x+\dfrac1x\right), x>0x > 0 is
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hard
Find the value of
log⁡5((55+5)(55+5)2(55+5)3⋯(6+25)215−18215−18215−18⋯3333)\log_{\sqrt5}\left(\frac{\sqrt{\left(5\sqrt5+5\right)\sqrt{\left(5\sqrt5+5\right)^2\sqrt{\left(5\sqrt5+5\right)^3\sqrt{\cdots}}}}} {\left(6+2\sqrt5\right)\sqrt[3]{215-18\sqrt[3]{215-18\sqrt[3]{215-18\sqrt[3]{\cdots}}}}}\right)
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hard
If x1x_1 and x2x_2 are the two solutions of the equation 3log⁡2x−12⋅xlog⁡169+27=03^{\log_2 x} - 12 \cdot x^{\log_{16} 9} + 27 = 0, then the value of x12+x22x_1^2 + x_2^2 is
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medium
Sum of all possible values of xx of the equation (3+2)x+(3−2)x−23=0(\sqrt{3}+\sqrt{2})^x + (\sqrt{3}-\sqrt{2})^x - 2\sqrt{3} = 0 is:
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medium
Find the number of negative integers satisfying −1≤x2−5x+4x2−4≤1-1 \leq \dfrac{x^2-5x+4}{x^2-4} \leq 1 is
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medium
Sum of all the roots of the equation log⁡7(2x−1)+log⁡7(2x−7)=1\log_7(2^x-1) + \log_7(2^x-7) = 1 is:
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medium
Find the number of single digit negative integers satisfying ∣∣x−6∣−∣x+6∣∣=12\big||x-6| - |x+6|\big| = 12 is
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medium
If the solution of the inequality 1<3x2−7x+8x2+1≤21 < \dfrac{3x^2-7x+8}{x^2+1} \leq 2 is [α,β][\alpha, \beta], then mark the incorrect option:
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medium
If SS is the set of all real xx such that x2(5−x)(1−2x)(5x+1)(x+2)\dfrac{x^2(5-x)(1-2x)}{(5x+1)(x+2)} is negative and 3x+16x3+x2−x\dfrac{3x+1}{6x^3+x^2-x} is positive, then SS contains:
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medium
The equation ∣x+1∣∣x−1∣=a2−2a−3|x+1||x-1| = a^2 - 2a - 3 can have real solutions for xx, if aa belongs to:
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medium
The solution set of the inequality 3x−4xx2−3x−4≥0\dfrac{3^x-4^x}{x^2-3x-4} \geq 0 is:
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medium
The complete set of values of xx for which the expression y=log⁡1/2(x−1x−2)y = \sqrt{\log_{1/2}\left(\dfrac{x-1}{x-2}\right)} is defined:
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medium
Number of integral solutions of the equation log⁡x−3(log⁡2x2−2x+3(x2+2x))=0\log_{x-3}\left(\log_{2x^2-2x+3}(x^2+2x)\right) = 0 is:
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medium
The complete solution set of the inequality 3x(2x−5)(x2+x+2)(cos⁡x−2)(x2+x)≤0\dfrac{3^x(2x-5)(x^2+x+2)}{(\cos x-2)(x^2+x)} \leq 0 is:
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medium
If 8x3+lx2−27x+m8x^3 + lx^2 - 27x + m is divisible by 2x2−x−62x^2 - x - 6, then l+ml + m is equal to:
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medium
If (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are all possible solutions of the system x2+y2+6x+2y=0x^2 + y^2 + 6x + 2y = 0 and x+y+8=0x + y + 8 = 0, then x12+x22+y12+y22x_1^2 + x_2^2 + y_1^2 + y_2^2 is equal to:
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medium
If 11+21=ab+cd\sqrt{11+\sqrt{21}} = \sqrt{\dfrac{a}{b}} + \sqrt{\dfrac{c}{d}}, where a,b,c,da, b, c, d are natural numbers with gcd⁡(a,b)=gcd⁡(c,d)=1\gcd(a,b) = \gcd(c,d) = 1, then a+b+c+da+b+c+d is equal to:
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medium
Real xx satisfying the equation 9log⁡3(log⁡2x)=log⁡2x−(log⁡2x)2+19^{\log_3(\log_2 x)} = \log_2 x - (\log_2 x)^2 + 1 is
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medium
Find the product of all solutions of the equation
34log⁡3x−3log⁡27x=2log⁡3x\frac{3}{4\log_3 x} - 3\log_{27} x = 2\log_3 x
 is
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medium
Number of solutions of the equation
log⁡3(3+x)+1log⁡(1+x2)33=1\log_3(3 + \sqrt{x}) + \frac{1}{\log_{(1+x^2)^3} 3} = 1
 is
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medium
Number of solutions of the equation
log⁡3(3+x)+1log⁡(1+x2)33=1\log_3(3 + \sqrt{x}) + \frac{1}{\log_{(1+x^2)^3} 3} = 1
 is
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medium
Number of values of xx satisfying ∣x2+4x+3∣+2x+5=0|x^2 + 4x + 3| + 2x + 5 = 0 is
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medium
Find the number of solutions of ∣x−4∣+3∣x+6∣−2x=5|x-4| + 3|x+6| - 2x = 5 is
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medium
Number of negative integer solutions of the equation ∣x+1∣−∣x∣+3∣x−1∣−2∣x−2∣=x+2|x+1| - |x| + 3|x-1| - 2|x-2| = x+2 is
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medium
Find the number of positive integers satisfying ∣x2−3x+2∣+∣x2+7x+12∣=10∣x+1∣|x^2-3x+2| + |x^2+7x+12| = 10|x+1| is
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medium
Find the number of positive integers satisfying ∣x2−1∣+∣x2−4∣≤3|x^2-1| + |x^2-4| \leq 3 is
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medium
Find the number of single digit prime numbers satisfying
(x−2)100(2+x)101(x−1)10x41(x+1)39(4−x)11≤0\frac{(x-2)^{100}(2+x)^{101}(x-1)^{10}}{x^{41}(x+1)^{39}(4-x)^{11}} \leq 0
 is
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medium
Find the sum of all values of xx satisfying the equation log⁡4(x−1)=log⁡2(x−3)\log_4(x-1) = \log_2(x-3) is
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medium
If pp denotes the product of the two values of xx satisfying the equation
log⁡2x(2)+log⁡4(2x)=−32\log_{2x}(2) + \log_4(2x) = -\frac{3}{2}
 then the value of log⁡2 ⁣(1p)\log_2\!\left(\dfrac{1}{p}\right) is
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medium
The equation log⁡x+1(x−0.5)=log⁡x−0.5(x+1)\log_{x+1}(x-0.5) = \log_{x-0.5}(x+1) has
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medium
Which of the following statements does not hold good
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medium
If ℓn(a+b3)=(ℓna+ℓnb2)\ell n\left(\dfrac{a+b}{3}\right) = \left(\dfrac{\ell na + \ell nb}{2}\right), then ab+ba\dfrac{a}{b} + \dfrac{b}{a} is equal to
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medium
Let mm, nn be real numbers satisfying the relation
log⁡2(m2+n2+1)+log⁡3(1+2sin⁡2(2π7)2−cos⁡(4π7))=log⁡2n+log⁡2(2m+2−n).\log_2\left(m^2+n^2+1\right)+\log_3\left(\frac{1+2\sin^2\left(\frac{2\pi}{7}\right)}{2-\cos\left(\frac{4\pi}{7}\right)}\right) = \log_2 n+\log_2\left(2m+2-n\right).
 Identify which of the following statement(s) is/are correct?
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medium
The solution set of the inequality
log⁡2x−1+12log⁡1/2x3+2>0\sqrt{\log_2x-1}+\frac12\log_{1/2}x^3+2>0
 may be
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easy
If x=∏n=12026nx = \prod_{n=1}^{2026} n, then the value of the expression
11log⁡2x+1log⁡3x+⋯+1log⁡2026x\frac{1}{\dfrac{1}{\log_2 x} + \dfrac{1}{\log_3 x} + \cdots + \dfrac{1}{\log_{2026} x}}
 is
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easy
Number of integral values of xx satisfying the inequality (34)6x+10−x2<2764\left(\dfrac{3}{4}\right)^{6x+10-x^2} < \dfrac{27}{64} is:
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easy
Number of natural numbers for which the number log⁡4−x(x2−14x+45)\log_{4-x}(x^2-14x+45) is defined is:
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easy
Sum of all the values of xx satisfying the equation log⁡17log⁡11(x+11+x)=0\log_{17}\log_{11}(\sqrt{x+11}+\sqrt{x}) = 0 is:
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easy
If ∣x∣−1∣x∣−2≥0\dfrac{|x|-1}{|x|-2} \geq 0, x∈Rx \in \mathbb{R}, then xx does not belong to:
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easy
If log⁡72x−62x−2>0\log_7\dfrac{2x-6}{2x-2} > 0, then x∈x \in:
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easy
The value of alog⁡ab−blog⁡baa^{\sqrt{\log_a b}} - b^{\sqrt{\log_b a}} (where a,b>0a, b > 0 and a≠1a \neq 1) is equal to
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easy
If 8x2+16x−51(2x−3)(x+4)>3\dfrac{8x^2+16x-51}{(2x-3)(x+4)} > 3, where xx is such that:
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easy
Number of integral values of xx which satisfies the inequality (x+6)2(x+5)(x+1)2(x−3)≤0\dfrac{(x+6)^2(x+5)(x+1)^2}{(x-3)} \leq 0 is:
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easy
The sum of all real roots of the equation ∣x−2∣2+∣x−2∣−2=0|x-2|^2 + |x-2| - 2 = 0 is
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easy
Find the number of solutions of ∣x−6∣+∣x+6∣<12|x-6| + |x+6| < 12 is
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easy
Find the number of single digit positive integers satisfying ∣x−5∣+∣x+5∣=2∣x∣|x-5| + |x+5| = 2|x| is
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easy
Let P(x)=x6+ax5+bx4+cx3+dx2+ex+fP(x) = x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f be a polynomial such that P(1)=1P(1)= 1, P(2)=2P(2) = 2, P(3)=3P(3) = 3, P(4)=4P(4) = 4, P(5)=5P(5) = 5 and P(6)=6P(6) = 6. Then the value of P(7)P(7) is equal to:
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