Logarithms and some other stuff

I have a few - a lot, actually - of doubts with logarithms and some other stuff. Anyone up? I'd be thankful even if you could answer one of these.

Q1. log5120+(x3)2log5(15x3)=log5(0.25x4) \log _{ 5 }{ 120 } +\left( x-3 \right) -2\cdot \log _{ 5 }{ \left( 1-{ 5 }^{ x-3 } \right) } =-\log _{ 5 }{ \left( 0.2-{ 5 }^{ x-4 } \right) }

Q2. Find the sum of all the solutions of the equation 3(log9x)292log9x+5=33\large{{ 3 }^{ { \left( \log _{ 9 }{ x } \right) }^{ 2 }-{ \frac { 9 }{ 2 } \log _{ 9 }{ x } }+5 }=3\sqrt { 3 } }

Q3. Let aa, bb, cc and dd, be positive integers such that logab=32\log_{a}{b}=\frac{3}{2} and logcd=54\log_{c}{d}=\frac{5}{4}. If (ac)=9(a-c)=9, find the value of (bd)(b-d).

Q4. If log3x45=log4x403\log_{3x}{45}=\log_{4x}{40\sqrt{3}} then find the characteristic of x3x^3 to the base 7.

Q5. Find xx satisfying the equation log2(1+4x)+log2(14x+4)=2log2(2x11)\log^{2}{\left(1+\frac{4}{x}\right)} + \log^{2}{\left(1-\frac{4}{x+4}\right)}=2\log^{2}{\left(\frac{2}{x-1}-1\right)}

Q6. Find the real solutions to the system of equations log10(2000xy)log10xlog10y=4\log_{10}{(2000xy)}-\log_{10}{x}\cdot\log_{10}{y}=4 log10(2yz)log10ylog10z=1\log_{10}{(2yz)}-\log_{10}{y}\cdot\log_{10}{z}=1 log10(zx)log10zlog10x=0\log_{10}{(zx)}-\log_{10}{z}\cdot\log_{10}{x}=0

Q7. Solve : log3(x+x1)=log9(4x3+4x1)\log _{ 3 }{ \left( \sqrt { x } +\left| \sqrt { x } -1 \right| \right) =\log _{ 9 }{ \left( 4\sqrt { x } -3+4\left| \sqrt { x } -1 \right| \right) } }

Q8. Prove that 2(logaab4+logbab4logaba4+logbab4)logab={22logab if ba>1 if 1<b<a\large{{ 2 }^{ \left( \sqrt { \log _{ a }{ \sqrt [ 4 ]{ ab } +\log _{ b }{ \sqrt [ 4 ]{ ab } } } } -\sqrt { \log _{ a }{ \sqrt [ 4 ]{ \frac { b }{ a } } } +\log _{ b }{ \sqrt [ 4 ]{ \frac { a }{ b } } } } \right) \cdot \sqrt { \log _{ a }{ b } } }=\begin{cases} 2 \\ { 2 }^{ \log _{ a }{ b } } \end{cases}\begin{matrix} { \text{ if }\quad b\geq a>1 } \\ \text{ if }\quad 1<b<a \end{matrix}}

Q9. Find the value of 1sin3α[sin3α+sin3(2π3+α)+sin3(4π3+α)]\frac { 1 }{ \sin { 3\alpha } } \left[ \sin ^{ 3 }{ \alpha } +\sin ^{ 3 }{ \left( \frac { 2\pi }{ 3 } +{ \alpha } \right) +\sin ^{ 3 }{ \left( \frac { 4\pi }{ 3 } +{ \alpha } \right) } } \right]

Q10. Find the value of aa for which the equation x24x+3=x+a\lvert x^2-4x+3\rvert = x+a has exactly three distinct real roots.

Q11. Find the number of terms of the longest geometric progression that can be obtained from the set (100, 101,, 1000)(100,~101,\dots ,~1000). (I think the question does not consider r=1r=1, because the answer given for this one is 66.)

Q12. If p(x)=ax2+bx+cp(x)=ax^2+bx+c and q(x)=ax2+dx+cq(x)=-ax^2+dx+c, where ac0ac\neq0, then prove that p(x)q(x)p(x)q(x) has at least two real roots. (I think it should be q(x)=ax2+bx+cq(x)=-ax^2+bx+c, but this is what the sheet says.)

Q13. If xx and yy are real numbers such that x2+2xyy2=6x^2+2xy-y^2=6 find the minimum value of (x2+y2)2\left(x^2+y^2\right)^2.

Q14. If the product (sin1)(sin3)(sin5)(sin7)(sin89)=12n(\sin1^{\circ})(\sin3^{\circ})(\sin5^{\circ})(\sin7^{\circ})\dots(\sin89^{\circ})=\frac{1}{2^n} then find the value of [n][n]. (where [y][y] denotes greatest integer less than or equal to yy.)

Note by Omkar Kulkarni
4 years, 2 months ago

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For Q11.

Should the answer be \infty? Let common ratio=1

Mehul Arora - 4 years, 2 months ago

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I believe the question doesn't consider that, because the answer given here is 66.

Omkar Kulkarni - 4 years, 2 months ago

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Please mention that. Otherwise other people may get confused like me :P

Mehul Arora - 4 years, 2 months ago

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@Mehul Arora Yeah, good idea :P

Omkar Kulkarni - 4 years, 2 months ago

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For 10, I get 2 solutions, a=1,a=0.75 a = -1, a = -0.75 .

Siddhartha Srivastava - 4 years, 2 months ago

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Could you explain how?

Omkar Kulkarni - 4 years, 2 months ago

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Uh, I did it graphically.

Draw x24x+3 x^2 - 4x + 3 ( Parabola cutting x -axis at (1,0), (3,0)). Invert the part under the x -axis to get the graph of x24x+3 |x^2 - 4x + 3| . Now look at the family of lines which have slope of 45 deg and see which of them cut the function at 3 points.

There should be two lines. One line cuts the graph at (1,0) and therefore must be y=x1 y = x - 1 .

The other is tangent to the graph between x =1 and x =3. The slope of the line is 1 1 . The slope of the function between x=1 x = 1 and x=3 x =3 is 2x+4 -2x +4 . Since the slopes are equal, x=3/2 x = 3/2 . Putting in the function y=3/4 y = 3/4 . Line which satisfies this is x=y0.75 x = y - 0.75 .

Siddhartha Srivastava - 4 years, 2 months ago

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@Siddhartha Srivastava Oh right. Thanks!

Omkar Kulkarni - 4 years, 2 months ago

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For question 14.

Pi Han Goh - 4 years, 2 months ago

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Thank you!

Omkar Kulkarni - 4 years, 2 months ago

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For question no. 1 , I got answer as x=1.

Anurag Pandey - 2 years, 9 months ago

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