CPhill

2log(x) = log(4x + 3)     note that we can write this as

log ( x² ) = log(4x + 3)     and equating expressions, we have

x² =  4x + 3                     rearranging, we have

x² - 4x - 3 = 0               no factoring is available here, so using the onsite solver, we have

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{3}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{2}}{\mathtt{\,-\,}}{\sqrt{{\mathtt{7}}}}\\
{\mathtt{x}} = {\sqrt{{\mathtt{7}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.645\: \!751\: \!311\: \!064\: \!590\: \!6}}\\
{\mathtt{x}} = {\mathtt{4.645\: \!751\: \!311\: \!064\: \!590\: \!6}}\\
\end{array} \right\}$$

Note something here...if we write this equation as 2log(x) = log(4x + 3), then we are explicitly assuming that x > 0  and only the positive solution is correct. However, if we write the equation as  log(x²) = log(4x + 3), then both solutions are correct  !!!

Wow, rosala....those pics and other info  are AMAZING !!!.....I'm going to sit here on this nice cool morning and ponder some of the info you have sent !!!

I wish you luck in your studies....if I had to do all of that now - at my "advanced" age - I think my brain would fall out of my head!!!  [ I 'd probably just sit next to you on test day and "peek" over at YOUR paper from time to time !!!  (LOL!!!) ]

I'll give you a whole garden of flowers for your nice presentation....!!!!

(BTW.......that's a picture from Butchart Gardens on Vancouver Island in Canada.....I visited there some years ago.....it's just as spectacular as the picture indicates !!!...it looks just like something that Dorothy would have seen when her house landed in Oz...)

That last inequality that Melody had to deal with IS tricky......here's a graph of the solution using Desmos:

Ah, DS.....your prior mistake has now been reversed by your correct answer, and your "future" has now been course-corrected........you return to where you were....... like it never even happened  !!!!

Relax, DS.....I doubt that  your mistake was enough to "alter" your future.....(but then again, you never know.......)

As "close" as you want.....[if you take calculus, this is one of its main tenets......we can always "approach" something very, very closely !!!  ]

A ratio of 5:3 means there were 8 parts of something. And 5 of the 8 parts said "yes." And 5/8ths of 240  = (5/8)*(240) = 5 * (240/8) = 5 * 30   = 150.  So, 150 people said "yes.'

a number is missing from the following list: 3, 5, -2, -4, x. (the missing value is x.) the mean of the list is -2. what is the value of x?

So we have

(3 + 5 + -2 + -4 + x)/5 = -2      multiply both sides by 5

3 + 5 + -2 + -4 + x  =  -10        simplify

2 + x   = -10                             subtract 2 from each side

x = -12

[(5 * x - 1) * 2 ] / 5 = 3     simplify

[(5x - 1 ) * 2] / 5  = 3    

(10x - 2) / 5  = 3       multiply by 5 on both sides

10x - 2  = 15             add 2 to both sides

10x = 17                   divide by 10 on both sides

x = 17/10

What is the answer to the square root of 3x plus 2 equals to 4?

If we have

√(3x + 2 ) = 4   square both sides

3x + 2   = 16     subtract 2 from both sides

3x = 14             divide by 3 on both sides

x = 14/3

If we have

√(3x)  + 2 = 4      subtract 2 from both sides

√(3x)  = 2             square both sides

3x = 4                  divide by 3 on both sides

x = 4/3