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This is the most extensive discussion on Logarithms and the number of ways that can be computed:

http://numbers.computation.free.fr/Constants/constants.html

i had a dream where someone punched me across the face demanding chicken eggs.

no but you are

TOO much

buy an IPad

eat my poopie little shpecial child

So how would you resolve this problem ?

\(\displaystyle \sqrt{-1}=\sqrt{\frac{1}{-1}}=\frac{\sqrt{1}}{\sqrt{-1}}=\frac{1}{\imath} \\ \displaystyle\sqrt{-1}=\sqrt{\frac{-1}{1}}=\frac{\sqrt{-1}}{\sqrt{1}}=\frac{\imath}{1}\)

Equating and cross multiplying,

\(\displaystyle \imath^{2}=1.\)

At what point has a mistake been made ?

-6% is -0.06, or a loss of 6%

No.  If \(x^2=-1\) then \(x=\pm \sqrt{-1}\) 

However \(\sqrt{-1} = i\)

Suppose that b  is a positive integer greater than or equal to 2. 

When 197 is converted to base b, the resulting representation has 4 digits.

What is the number of possible values for b?

\(\begin{array}{|rcll|} \hline 197_{10} &=& 11000101_2 \\ 197_{10} &=& 21022_3 \\ 197_{10} &=& 3011_{\color{red}4} \\ 197_{10} &=& 1242_{\color{red}5} \\ 197_{10} &=& 525_6 \\ 197_{10} &=& 401_7 \\ 197_{10} &=& 305_8 \\ 197_{10} &=& 238_9 \\ 197_{10} &=& 197_{10} \\ \cdots \\ \hline \end{array}\)

b is 4 or 5

123.8 IS a decimal     123 8/10   or 123 4/5 as a fraction.

If sin(A) = -0.4382.  0 < A < 360.

Find two possible values for A.

\(\begin{array}{|rcll|} \hline \sin(A) &=& -0.4382 \\ A_1 &=& \arcsin(-0.4382) + z\cdot 360^{\circ} \quad & | \quad z \in \mathbb{Z} \\ A_1 &=& -25.9890903445^{\circ} + z\cdot 360^{\circ} \\ A_1 &=& -25.9890903445^{\circ}+360^{\circ} + z\cdot 360^{\circ} \\ \mathbf{A_1} &\mathbf{=}& \mathbf{334.010909656^{\circ} + z\cdot 360^{\circ}} \\\\ \sin(A)=\sin(180^{\circ}-A) &=&-0.4382 \\ 180^{\circ}-A_2 &=& \arcsin(-0.4382) + z\cdot 360^{\circ} \quad & | \quad z \in Z \\ A_2 &=& 180^{\circ}- \arcsin(-0.4382) + z\cdot 360^{\circ} \\ A_2 &=& 180^{\circ}- (-25.9890903445^{\circ}) + z\cdot 360^{\circ} \\ A_2 &=& 180^{\circ}+25.9890903445^{\circ}) + z\cdot 360^{\circ} \\ \mathbf{A_2} &\mathbf{=}&\mathbf{205.989090344^{\circ}+ z\cdot 360^{\circ}} \\ \hline \end{array} \)

0 < A < 360.

\(\begin{array}{|rcll|} \hline \mathbf{A_1} &\mathbf{=}& \mathbf{334.010909656^{\circ} } \\ \mathbf{A_2} &\mathbf{=}& \mathbf{205.989090344^{\circ}} \\ \hline \end{array}\)

Arcsin  -  .4382 =  334.01 degrees      205.98 degrees

Every number has two square roots.

\(\displaystyle \sqrt{-1}= \pm\imath\)

If you choose to say that f(0) = i,

then you are making a choice.

Ditto with g(x).

The following article might be of some use:

http://mathforum.org/library/drmath/view/55563.html

.

v20 = 1/4*(v20 + 1)

v20 = 1/4*v20 + 1/4

v20 = 1/3 m/s

1/4*v₁₀² = v₂₀²

1/4*v₁₀² = 1/9

v₁₀² = 4/9

v10 = 2/3 m/s

Son's starting speed = 1/3 m/s

Father's starting speed = 2/3 m/s

I have no idea why the answers are different because I can't follow your logic well, but atleast this part of mine was wrong

(1/2)M(V+1)² = (1/2)mv²  So (V+1)² = v2/2

Instead it has to cancel out the M and m by (1/2)M(V+1)² = (1/2)mv²  So (V+1)² = v²/4

that should account for the discrepancies not the velocity thing I said before

No, it assumes the father's starting speed is V and the son's is v.  It also assumes the son's speed doesn't change when the father's does (indeed, it assumes the son's speed doesn't change at any time).

.

This assumes the father and son have the same start velocity.

A line

log₁₃₆(680) = 1.3276106198925393

K = 1/2*mv²

K₁₀  = 2 * K₂₀

^{K₁₀ = K_2F}

m₁ = 1/2 m₂

K₁₀ = 1/4*m₂v²

K₂₀ = 1/2*m₂*v²

1/4*m₂v₁₀² = 2 * 1/2*m₂v₂₀²

1/4*v₁₀² = v₂₀²

^K_2F = 1/2*m₂*(v₂₀ + 1)²

1/4*m₂*v₁₀² = 1/2*m₂*(v₂₀ + 1)² 

1/4*v₁₀² = 1/2*(v₂₀ + 1)² 

1/4*v₁₀² = v₂₀² 

v₂₀² = 1/2*(v₂₀ + 1)² 

v_{20 =} 1/2*(v₂₀ + 1)

v₂₀ = 1/2_*v₂₀ + 1/2

v_{20 = 1 m/s}

1/4*v₁₀² = v₂₀²

1/4*v₁₀² = 1

v₁₀2 = 4

v₁₀ = 2 m/s

Son's starting speed = 2 m/s

Father's starting speed = 1 m/s

Upper case = father, lower case = son

(1/2)MV² = (1/2)*(1/2)mv²

m = M/2.  So. V² = v²/4.   

(1/2)M(V+1)² = (1/2)mv²  So (V+1)² = v²/2, or  V² + 2V + 1 = v²/2

So. V² + 2V + 1 = 4V²/2. 

V² - 2V - 1 = 0

V² - 2V + 1 = 2

(V-1)² = 2

V = 1 + √2.   (Since only the positive V is needed) This is the father's original speed (in m/s)

Use the fact that v² = 4V² to find the son's speed.

If 25.12 is the area of a circle, what is the circumference?

Let A = 25.12 the area of a circle

Let C = circumference of a circle

1. Radius r = ?

\(\begin{array}{|rcll|} \hline A &=& \pi r^2 \\ \frac{A}{\pi} &=& r^2 \\ \mathbf{\sqrt{\frac{A}{\pi} } } & \mathbf{=} & \mathbf{r} \\ \hline \end{array}\)

2. Circumference

\(\begin{array}{|rcll|} \hline C &=& 2\pi r \quad &| \quad r=\sqrt{\frac{A}{\pi}} \\ C &=& 2\pi\sqrt{\frac{A}{\pi}} \\ C &=& 2\sqrt{\pi^2\frac{A}{\pi}} \\ \mathbf{C }& \mathbf{=} & \mathbf{2\sqrt{A\cdot \pi }} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline \mathbf{C }& \mathbf{=} & \mathbf{2\sqrt{A\cdot \pi }} \quad & | \quad A = 25.12 \\ C &=& 2\sqrt{25.12\cdot 3.14159265359} \\ C &=& 2\sqrt{78.9168074582} \\ C &=& 2\cdot 8.88351323848 \\ C &=& 17.7670264770 \\ \hline \end{array}\)