# 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

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
4 years, 5 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

For Q11.

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

- 4 years, 5 months ago

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

- 4 years, 5 months ago

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

- 4 years, 5 months ago

Yeah, good idea :P

- 4 years, 5 months ago

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

- 4 years, 5 months ago

Could you explain how?

- 4 years, 5 months ago

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 $x = y - 0.75$.

- 4 years, 5 months ago

Oh right. Thanks!

- 4 years, 5 months ago

- 4 years, 5 months ago

Thank you!

- 4 years, 5 months ago

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

- 3 years ago