CPhill

First, divide both sides by 9....this gives...

SinA = (.6/9) = (.54)

Using the sine inverse function, we can find A

asin(.54) = 32.683638846258°    But since you specified that this angle lies in the second quadrant, we have 180°-32.683638846258° = 147.316361153742°

Let the breadth be B. Then, since the length is twice the breadth, call the length 28. And the perimeter of any rectangle is given by P = 2(W + L) where W is the width (breadth) and L is the length. So we have:

108 = 2(B + 2B)     Divide by 2

54 = B + 2B            Simplify on the right

54 = 3B                  Divide by 3

18 = B                    And that's the width (breadth)      And the length is twice that = 2(18) = 36

Check that 36 + 36 + 18 + 18 = 108    .....Yep !!!!

I assume you mean this:   log₆ 45

A log is an exponent...in this case it's the exponent (power) - call it "x" - that we put on 6 to raise it to 45. We can evaluate this by something called the "change of base" rule, Specifically:

x = log 45 / log6   ≈ 2.1245387872348441

Verify for yourself that 6^{2.1245387872348441} ≈ 45

Multiply everything through by 100 to clear the decimals.

26x^2 - 105x - 100 = 0

This doesn't look "factorable".........let's use the on-site solver. (And the quadratic formula!!)......

$${\mathtt{26}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{105}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{100}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\mathtt{5}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{857}}}}{\mathtt{\,-\,}}{\mathtt{105}}\right)}{{\mathtt{52}}}}\\
{\mathtt{x}} = {\frac{\left({\mathtt{5}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{857}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{105}}\right)}{{\mathtt{52}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.795\: \!630\: \!993\: \!904\: \!701\: \!4}}\\
{\mathtt{x}} = {\mathtt{4.834\: \!092\: \!532\: \!366\: \!239\: \!8}}\\
\end{array} \right\}$$

And there are the two solutions .....  !!!

Good job, rosala....I have some sunglasses to share!!

OK, subtract 2x from both sides....this gives

2/y = 2x + 2       divide everything by 2....we have

1/y = x +1          subtract 1 from both sides

1/y - 1  = x         which we could write as

(1 - y)/y  = x

And that's it....

Good job, snip3s and nice graphic ND......points and thumbs for both...!!!

BTW, ND......how DO you generate graphics like that??....very impressive....!!!

Add 18 to both sides....this gives

x/5 = 2     multiply both sides by 5

x = 10

And that's it......!!!!

I assume you mean this:

log₃ 7         ......  We can use the "Change of Base" rule to evaluate this....it says that the "power" - let's call it "x" - needed to raise 3 to 7 is given by:

x = log(7)/log(3)  ≈  1.7712437491614222

Check for yourself that 3^{1.7712437491614222}  ≈ 7