Properties of Logarithm

1. If   $y = a^x$,   then   $\log_a y = x$.   ← Definition of logarithm
2. $\log_a xy = \log_a x + \log_a y$
3. $\log_a \dfrac{x}{y} = \log_a x - \log_a y$
4. $\log_a x^n = n \log_a x$
5. $\log_a a = 1$
6. $\log_a 1 = 0$
7. $\log_{10} x = \log x$   ←   Common logarithm
8. $\log_e x = \ln x$   ←   Naperian or natural logarithm
9. $\log_y x = \dfrac{\log x}{\log y} = \dfrac{\ln x}{\ln y}$   ←   Change base rule
10. If   $\log_a x = \log_a y$,   then   $x = y$.
11. If   $\log_a x = y$,   then   $x = {\rm antilog}_a \, y$.

Laws of Exponents (Index Law)
1. $x^n = x \cdot x \cdot x ... \, (n \text{ factors})$

2. $x^m \cdot x^n = x^{m + n}$

3. $(x^m)^n = x^{mn}$

4. $(xyz)^n = x^n \, y^n \, z^n$

5. $\dfrac{x^m}{x^n} = x^{m - n}$

6. $\left( \dfrac{x}{y} \right)^n = \dfrac{x^n}{y^n}$

7. $x^{-n} = \dfrac{1}{x^n}$   and   $\dfrac{1}{x^{-n}} = x^n$

8. $x^0 = 1$,   provided   $x \ne 0$.

9. $(x^m)^{1/n} = (x^{1/n})^m = x^{m/n}$

10. $x^{m/n} = \sqrt[n]{x^m}$

11. If   $x^m = x^n$,   then   $m = n$   provided   $x \ne 0$.
 

Problem 338
The two 12-ft beams shown in Fig. 3-16 are to be moved horizontally with respect to each other and load P shifted to a new position on CD so that all three reactions are equal. How far apart will R₂ and R₃ then be? How far will P be from D?
 

Problem 337
The upper beam in Fig. P-337 is supported at D and a roller at C which separates the upper and lower beams. Determine the values of the reactions at A, B, C, and D. Neglect the weight of the beams.
 

Problem 336
The cantilever beam shown in Fig. P-336 is built into a wall 2 ft thick so that it rests against points A and B. The beam is 12 ft long and weighs 100 lb per ft.
 

Problem 335
The roof truss in Fig. P-335 is supported by a roller at A and a hinge at B. Find the values of the reactions.
 

Problem 334
Determine the reactions for the beam loaded as shown in Fig. P-334.
 

Problem 333
Determine the reactions R₁ and R₂ of the beam in Fig. P-333 loaded with a concentrated load of 1600 lb and a load varying from zero to an intensity of 400 lb per ft.
 

Problem 332
Determine the reactions for the beam shown in Fig. P-332.
 

Problem 329
Two cylinders A and B, weighing 100 lb and 200 lb respectively, are connected by a rigid rod curved parallel to the smooth cylindrical surface shown in Fig. P-329. Determine the angles α and β that define the position of equilibrium.