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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. \[ \log _{ 5 }{ 120 } +\left( x-3 \right) -2\cdot \log _{ 5 }{ \left( 1-{ 5 }^{ x-3 } \right) } =-\log _{ 5 }{ \left( 0.2-{ 5 }^{ x-4 } \right) } \]

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

Q3. Let \(a\), \(b\), \(c\) and \(d\), be positive integers such that \(\log_{a}{b}=\frac{3}{2}\) and \(\log_{c}{d}=\frac{5}{4}\). If \((a-c)=9\), find the value of \((b-d)\).

Q4. If \[\log_{3x}{45}=\log_{4x}{40\sqrt{3}}\] then find the characteristic of \(x^3\) to the base 7.

Q5. Find \(x\) satisfying the equation \[\log^{2}{\left(1+\frac{4}{x}\right)} + \log^{2}{\left(1-\frac{4}{x+4}\right)}=2\log^{2}{\left(\frac{2}{x-1}-1\right)}\]

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

Q7. Solve : \[\log _{ 3 }{ \left( \sqrt { x } +\left| \sqrt { x } -1 \right| \right) =\log _{ 9 }{ \left( 4\sqrt { x } -3+4\left| \sqrt { x } -1 \right| \right) } } \]

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

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

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

Q11. Find the number of terms of the longest geometric progression that can be obtained from the set \((100,~101,\dots ,~1000)\). (I think the question does not consider \(r=1\), because the answer given for this one is \(6\).)

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

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

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

Note by Omkar Kulkarni
2 years, 1 month ago

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

Anurag Pandey - 8 months, 2 weeks ago

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For question 14.

Pi Han Goh - 2 years, 1 month ago

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Thank you!

Omkar Kulkarni - 2 years, 1 month ago

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

Siddhartha Srivastava - 2 years, 1 month ago

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Could you explain how?

Omkar Kulkarni - 2 years, 1 month ago

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Uh, I did it graphically.

Draw \( x^2 - 4x + 3 \) ( Parabola cutting x -axis at (1,0), (3,0)). Invert the part under the x -axis to get the graph of \( |x^2 - 4x + 3| \). 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 \( y = x - 1 \).

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

Siddhartha Srivastava - 2 years, 1 month ago

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@Siddhartha Srivastava Oh right. Thanks!

Omkar Kulkarni - 2 years, 1 month ago

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For Q11.

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

Mehul Arora - 2 years, 1 month ago

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I believe the question doesn't consider that, because the answer given here is \(6\).

Omkar Kulkarni - 2 years, 1 month ago

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

Mehul Arora - 2 years, 1 month ago

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@Mehul Arora Yeah, good idea :P

Omkar Kulkarni - 2 years, 1 month ago

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