5 5. 5.. a a n n A m n a m a n = a m+n (a m ) n = a mn 3 (ab) n = a n b n a n n 0 3 3 0 = 3 +0 = 3, 3 3 = 3 +( ) = 3 0 3 0 3 3 0 = 3 3 =, 3 = 30 3 = 3 0 a 0 a`n a 0 n a 0 = a`n = a n a` = a 83
84 5 5. 0 = 0 3 = 0 = 3 000 34.5 0 3 = 34.5 0.00 = 0.0345 0.03 =.3 0.0 =.3 0 ( ) 4 = = 4 4 5. (3) () 4 = () ( ) 5 = (3) 3. 0 4 = (4) 0.00074 = 7.4 0 3 5 3 = 3 5 3 = 33 = 3 5+( ), 3 5 3 = 35 3 = 35 3 = 3 5+ = 3 5 ( ) (3 5 ) = (3 5 ) = 3 5 = 3 (5 ) = 3 5 ( ) (3 5) = (3 5) = 3 5 = 3 5 = 3 5 ( ) a 0 b 0 m n a m a n = a m+n am a n = am`n 3 (a m ) n = a mn 4 (ab) n = a n b n 5. () 3 4 3 = 3 () 0 3 0 = 0 (3) (3 ) 4 = 3 (4) ( 3 4 ) = 3
5.. 85 B n n a a n n = a a n a 3 4 a 5. () 3 = 8 8 3 () 3 4 = ( 3) 4 = 8 3 3 8 4 5.3 () ( ) 3 = 8 8 () 4 = ( ) 4 = 6 6 a n y = n ( 0) y y = n ( 0) a n = a a n a a a O n a 5.3 () 3 = 8 3 8 = () 3 4 = 8 4 8 = 3 a > 0 n a > 0, ( n a) n = a n an = a 5.4 () 3 = () 3 7 = (3) 4 6 =
86 5 n a a > 0 b > 0 m n n a n b = n n a a ab n = n b b 3 ( n a) m = n a m 4 m n a = mn a 3 (( n a) m ) n = (( n a) n ) m = a m n a > 0 ( n a) m > 0 ( n a) m = n a m n a m n a m 4 5.4 () () 3 3 4 = 3 4 = 3 3 = 3 3 3 = 3 3 = 3 4 (3) ( 4 5) 3 = 4 5 3 = 4 5 3 (4) 3 = 3 3 = 6 3 5.5 () 3 4 3 3 3 9 () 4 (3) ( 4 5) (4) 4 3
5.. 87 C a > 0 a (a ) = a (a ) = a a = a = a a > 0 m n r a n = n a, a m n = ( n a) m = n a m, a`r = a r 5.5 4 = 4 3 3 = 3 3 = 3 9 5 = 5 = 5 = 5 5.6 5.5 () 4 3 () 3 3 4 (3) 5 3 (4) 3 ( ) a > 0 b > 0 r s a r a s = a r+s ar a s = ar`s 3 (a r ) s = a rs 4 (ab) r = a r b r r = 3 s = 3 a 3 a = a 4 6 a 3 6 = 6 a 4 6 a 3 = 6 a 7 = a 7 6 = a 3 + (a 3 ) = 3 a = 6 a = a 6 = a 3
88 5 5. () 8 8 3 8 6 () 3 6 () 8 8 3 8 6 = 8 + 3 6 = 8 3 = ( 3 ) 3 = 3 3 = = 4 () 3 6 = 3 6 = + 3 + 6 = = 5.7 () 64 3 6 4 () 4 9 6 7 3 9 a > 0 a r r =.44.4,.4,.44,.44, r s n n a n y n = a n a y = n 3 8 = 3 ( ) 3 = 5 = 5 ( ) 5 = n a O n n 0 a n = a a
5.. 89 5.. a y = a a A y = a y = y = = 4 = 0.5 0.5 = = = 0.7.5 = 3 = =.83 y = O 5.8.5 0.5 0 0.5.5 y 0.5 0.7.83 4 ( ) y = y y = y y = f() y = f( ) y O y = ( )
90 5 y = a a > y y 0 < a < a a O O (0, ) (, a) a > 0 < a < 5.9 ( ) () y = 3 () y = 3 B y = a y = a a > y r < s a r < a s 3 0 < a < y r < s a r > a s a > 0 a r = s a r = a s y y
5.. 9 5.6 y = y = ( 3 < 0 <.5 ) ( ) 3 > ( ) 0 > ( ).5 5.0 3 () 3 0.5 3 () ( ) 3 ( 3 3 ) 4 ( 3 ) C 5. ( ) () 4 = 8 () 8 (3) < 3 9 () = 3 4 =( ) = = 3 = 3 () 3 y= 3 ( ) ( ) (3) < 3 3 > y=( 3)
9 5 5. ( ) () = 6 () = 6 (3) 3 = 9 (4) 8 = 4 (5) 3 8 (6) ( ) > 3 5..3 3.0 0 8 m km 3.3 0
5.. 93 () ( 4 3 + 4 )( 4 3 4 ) () (8 3 8 3 )(8 3 + + 8 3 ) 3 () y = () y = + 4 () ( ) = 8 () 7 = 3 (3) ( ) + 3 ( ) 9
94 5 6 km 0 3 (3.0 0 8 ) () 3 () 63 8 3 () () y y 3 O O 4 () = 3 () = (3)
5.. 95 5. 5.. y = y A y y = y = q q = q log q 5.7 () 8 = 3 log 8 = 3 () = log = O log q 5. () log 6 = () log 4 = (3) log = 0 y = a y = a M M = a p p p log a M a M log a M M y M 0 < a < log logarithm log a M O a > 0 a M > 0 M = a p log a M = p log a M a > 0 a M > 0
96 5 5.8 () 8 = 3 4 log 3 8 = 4 () log 4 6 = 6 = 4 6=4 4 = 4 = log 4 6 = 5.3 () 9 = 3 log 3 9 = () 5 = 5 log 5 5 = (3) 3 = 4 = log 3 (4) log 4 64 = = log a M = p M = a p log a a p = p 5.9 () log 5 5 = log 5 5 3 = 3 log a M M a p () log 3 3 = log3 3 = 5.4 () log 5 () log 5 5 (3) log 3 7 (4) log 6 (5) log 0 0. (6) log 5 5 (7) log 3 (8) log 3 3
5.. 97 B a = a 0 a = a log a = 0 log a a = M > 0 N > 0 k log a MN = log a M + log a N log a M N = log a M log a N 3 log a M k = k log a M log a M = p log a N = q log a N = log a N M = a p, N = a q MN = a p a q = a p+q log a MN = p + q = log a M + log a N 5.5 3 log a M = p log a N = q M = a p, N = a q M N = ap a q = ap q M log a N = p q = log a M log a N 3 log a M = p M = a p M k = (a p ) k = a kp log a M k = kp = k log a M
98 5 5.0 () log 0 + log 0 5 = log 0 ( 5) = log 0 0 = 4 () log 4 log 3 = log 3 3 = log 8 = log 3 = 3 (3) log 3 4 + log 3 5 log 3 8 = log 3 4 + log 3 5 log 3 8 6 5 = log 3 = log 8 3 0 5.6 () log 4 + log 4 8 () log 3 log 3 8 (3) log 3 4 + log 3 8 3 log 3 (4) log 0 5 log 0 5 + log 0 9 C a log a b c log a b = p b = a p log c b = log c a p log c b = p log c a a log c a 0 p = log c b log c a a b c a c log a b = log c b log c a
5.. 99 5. () log 8 6 8 = 3 6 = 4 log 8 6 = log 6 log 8 = log 4 log 3 = 4 3 () log 3 log 3 8 log a log b log 3 log 3 8 = log 3 log 8 log 3 = log 8 = log 3 = 3 5.7 () log 4 8 () log 9 3 (3) log 3 log 7 5.. a y = log a a A y = log y = log = y 89 y = y 0.5 0.35 0.5 0.7.4.83 4 y.5 0.5 0 0.5.5 y y (, y) 3 O 3 4 5 6 7 8 9 y = log 3
00 5 y y = Q(t, s) y =. t = log s s = t P(s, t) O Q(t, s) y = P(s, t). P(s, t) Q(t, s) y = y = log a y = a y = y a > y 0 < a < y = a O a y = a O a y = y = log a y = y = log a y (, 0) (a, ) a > 0 < a < 5.8 () y = log 3 () y = log
5.. 0 B y = log a y = log a a > 0 < p < q log a p < log a q 3 0 < a < 0 < p < q log a p > log a q p > 0 q > 0 p = q log a p = log a q 5. y = log y 3 log 3 y log log 3 = log 8 3 y = log > 0 O 0 < 8 y = log y = 3 8 5.9 y = log y 4 y = log y
0 5 C 5. () log + log ( ) = 3 () log ( ) > () M > 0 N > 0 log a M + log a N = log a MN () > 0 > 0 > log ( ) = 3 ( ) = 3 ( + )( 4) = 0 + > 0 = 4 () log ( ) > log ( ) 4=0 > 0 < 4 < < 5 4 y=log ( ) 5.0 () log ( + 5) + log ( ) = 3 () log ( + ) 3 3
5.. 03 5..3 60000 =.6 0 6, 0.00068 = 6.8 0 4 0 n 0 A 0 M M = a 0 n n a < 0 log 0 M n log 0 a log 0 M = log 0 a + log 0 0 n = log 0 a + n a.00.0.0 9.99 log 0 a 5 4 5.3 0 3.0.0000.0043.0086.08 log 0.6 = 0.095..044.0453.049.053 log 0 60000 = log 0 (.6 0 6 )..079.088.0864.0899 = log 0.6 + log 0 0 6.3.39.73.06.39.4.46.49.53.533 = 0.095 + 6 = 6.095.5.76.790.88.847 log 0 0.006 = log 0 (.6 0 3 ).6.04.068.095..7.304.330.355.380 = log 0.6 + log 0 0 3.8.553.577.60.65.9.788.80.833.856 = 0.095 3 =.7905.0.300.303.3054.3075 5. 4 () log 0 3450 () log 0 9000 (3) log 0 0.00068 a < 0 log 0 a 0 log 0 a <
04 5 B N log 0 N N 3 N 00 N < 000 0 N < 0 3 log 0 0 log 0 N < log 0 0 3 log 0 N < 3 n 0 n 5N<0 n n 5log 0 N<n log 0 N < 3 N 3 5. N log 0 N () () 5 (3) 0 5.3 3 0 log 0 3 = 0.477 log 0 3 0 = 0 log 0 3 = 0 0.477 = 9.54 9 < log 0 3 0 < 0 log 0 0 9 < log 0 3 0 < log 0 0 0 0 9 < 3 0 < 0 0 3 0 0 5.3 log 0 = 0.300 () 0 () 30
5.. 05 5. log 0 = 0.300 n 0 n n 0 0 9 n < 0 0 n n 0 0 9 n < 0 0 9 n log 0 < 0 9 log 0 n < 0 log 0 log 0 = 0.300 9 log 0 = 9 0.300 = 9.9 0 log 0 = 0 0.300 = 33. n n = 30, 3, 3, 33 5.4 log 0 3 = 0.477 3 n 8 n
06 5 0 < M < M log 0 M M 3 0 M 0.00 M < 0.0 0 3 M < 0 3 log 0 M < 3 log 0 M < M 3 0 5.5 0 < M < M 5 0 log 0 M 5..4 5 () log 4 8 + log 4 () log 3 3 log 3
6 y = log ( ) 5.. 07 7 () log 3 ( + ) = () log ( ) log 8 log 0 = 0.300 () log 0 5 () 0. 0 0
08 5 5 () () 3 6 y O 3 7 () =, 4 () 0 < 8 () 0.6990 () 7 5.3 5.3. A () 36.5 3 0. () 9 4 9 3 9 () 8 () 0.5 4
5.3. 09 3 () 3 + = 3 ( ) 9 () 8 4 + (3) ( ) 4 () log 5 3+3 log 5 log5 4 () (log 3 + log 4 9)(log 3 4 + log 9 ) 5 log 0 = a log 0 3 = b a b 3 () log 0 () log 3 8 0 6 (3) log 3 (4) log 0 5
0 5 6 a b c log a b log b c log c a = 7 () log 0.5 ( + )( + ) = () log 3 ( ) + log 3 ( 7) = 8 log 0 = 0.300 30 00
5.3. 5.3. B 9 () y = + ( 3 3) () y = log ( + ) (0 ) 0 4 3 4 = 0 () = t t () () y = 4 () y = (log 3 ) log 3
5 () log 0.5 (3 ) log 0.5 () log ( + ) + log ( ) < 3 M = 3 5 () log 0 M () M
5.3. 3 ( ) n 4 log 0 = 0.300 < 0 n 4 5 log 0 = 0.300 log 0 3 = 0.477 n < 3 0 < n+ n 0 () > 0 t > 0 () log 3 = t 3 () log 0 M 4 log 0 = log 0 5 3 0 =
4 5 () 08 () 3 () 3 () 0 3 () = 3 () (3) 3 4 () 0 () 5 5 () b 3a () 3 (a + b) (3) b a [ 6 = log b log a c b log ] a a a log a log a c (4) a + b 7 () = 0, 3 () = 5 8 598 30n 00 n > 0 6 9 () 4 y 6 () 3 y 0 () t 3t 4 = 0 () = () = () = 3 4 () < 3 () < < 3 3 () 0.330 ().7 [ ] 4 n = 4 n log 0 < 4 5 n = 3
5.4 () 0 3 4 5 6 7 8 9.0.0000.0043.0086.08.070.0.053.094.0334.0374..044.0453.049.053.0569.0607.0645.068.079.0755..079.088.0864.0899.0934.0969.004.038.07.06.3.39.73.06.39.7.303.335.367.399.430.4.46.49.53.553.584.64.644.673.703.73.5.76.790.88.847.875.903.93.959.987.04.6.04.068.095..48.75.0.7.53.79.7.304.330.355.380.405.430.455.480.504.59.8.553.577.60.65.648.67.695.78.74.765.9.788.80.833.856.878.900.93.945.967.989.0.300.303.3054.3075.3096.38.339.360.38.30..3.343.363.384.3304.334.3345.3365.3385.3404..344.3444.3464.3483.350.35.354.3560.3579.3598.3.367.3636.3655.3674.369.37.379.3747.3766.3784.4.380.380.3838.3856.3874.389.3909.399.3945.396.5.3979.3997.404.403.4048.4065.408.4099.46.433.6.450.466.483.400.46.43.449.465.48.498.7.434.4330.4346.436.4378.4393.4409.445.4440.4456.8.447.4487.450.458.4533.4548.4564.4579.4594.4609.9.464.4639.4654.4669.4683.4698.473.478.474.4757 3.0.477.4786.4800.484.489.4843.4857.487.4886.4900 3..494.498.494.4955.4969.4983.4997.50.504.5038 3..505.5065.5079.509.505.59.53.545.559.57 3.3.585.598.5.54.537.550.563.576.589.530 3.4.535.538.5340.5353.5366.5378.539.5403.546.548 3.5.544.5453.5465.5478.5490.550.554.557.5539.555 3.6.5563.5575.5587.5599.56.563.5635.5647.5658.5670 3.7.568.5694.5705.577.579.5740.575.5763.5775.5786 3.8.5798.5809.58.583.5843.5855.5866.5877.5888.5899 3.9.59.59.5933.5944.5955.5966.5977.5988.5999.600 4.0.60.603.604.6053.6064.6075.6085.6096.607.67 4..68.638.649.660.670.680.69.60.6.6 4..63.643.653.663.674.684.694.6304.634.635 4.3.6335.6345.6355.6365.6375.6385.6395.6405.645.645 4.4.6435.6444.6454.6464.6474.6484.6493.6503.653.65 4.5.653.654.655.656.657.6580.6590.6599.6609.668 4.6.668.6637.6646.6656.6665.6675.6684.6693.670.67 4.7.67.6730.6739.6749.6758.6767.6776.6785.6794.6803 4.8.68.68.6830.6839.6848.6857.6866.6875.6884.6893 4.9.690.69.690.698.6937.6946.6955.6964.697.698 5.0.6990.6998.7007.706.704.7033.704.7050.7059.7067 5..7076.7084.7093.70.70.78.76.735.743.75 5..760.768.777.785.793.70.70.78.76.735 5.3.743.75.759.767.775.784.79.7300.7308.736 5.4.734.733.7340.7348.7356.7364.737.7380.7388.7396 5
5.5 () 0 3 4 5 6 7 8 9 5.5.7404.74.749.747.7435.7443.745.7459.7466.7474 5.6.748.7490.7497.7505.753.750.758.7536.7543.755 5.7.7559.7566.7574.758.7589.7597.7604.76.769.767 5.8.7634.764.7649.7657.7664.767.7679.7686.7694.770 5.9.7709.776.773.773.7738.7745.775.7760.7767.7774 6.0.778.7789.7796.7803.780.788.785.783.7839.7846 6..7853.7860.7868.7875.788.7889.7896.7903.790.797 6..794.793.7938.7945.795.7959.7966.7973.7980.7987 6.3.7993.8000.8007.804.80.808.8035.804.8048.8055 6.4.806.8069.8075.808.8089.8096.80.809.86.8 6.5.89.836.84.849.856.86.869.876.88.889 6.6.895.80.809.85.8.88.835.84.848.854 6.7.86.867.874.880.887.893.899.8306.83.839 6.8.835.833.8338.8344.835.8357.8363.8370.8376.838 6.9.8388.8395.840.8407.844.840.846.843.8439.8445 7.0.845.8457.8463.8470.8476.848.8488.8494.8500.8506 7..853.859.855.853.8537.8543.8549.8555.856.8567 7..8573.8579.8585.859.8597.8603.8609.865.86.867 7.3.8633.8639.8645.865.8657.8663.8669.8675.868.8686 7.4.869.8698.8704.870.876.87.877.8733.8739.8745 7.5.875.8756.876.8768.8774.8779.8785.879.8797.880 7.6.8808.884.880.885.883.8837.884.8848.8854.8859 7.7.8865.887.8876.888.8887.8893.8899.8904.890.895 7.8.89.897.893.8938.8943.8949.8954.8960.8965.897 7.9.8976.898.8987.8993.8998.9004.9009.905.900.905 8.0.903.9036.904.9047.9053.9058.9063.9069.9074.9079 8..9085.9090.9096.90.906.9.97.9.98.933 8..938.943.949.954.959.965.970.975.980.986 8.3.99.996.90.906.9.97.9.97.93.938 8.4.943.948.953.958.963.969.974.979.984.989 8.5.994.999.9304.9309.935.930.935.9330.9335.9340 8.6.9345.9350.9355.9360.9365.9370.9375.9380.9385.9390 8.7.9395.9400.9405.940.945.940.945.9430.9435.9440 8.8.9445.9450.9455.9460.9465.9469.9474.9479.9484.9489 8.9.9494.9499.9504.9509.953.958.653.958.9533.9538 9.0.954.9547.955.9557.956.9566.957.9576.958.9586 9..9590.9595.9600.9605.9609.964.969.964.968.9633 9..9638.9643.9647.965.9657.966.9666.967.9675.9680 9.3.9685.9689.9694.9699.9703.9708.973.977.97.977 9.4.973.9736.974.9745.9750.9754.9759.9763.9768.9773 9.5.9777.978.9786.979.9795.9800.9805.9809.984.988 9.6.983.987.983.9836.984.9845.9850.9854.9859.9863 9.7.9868.987.9877.988.9886.9890.9894.9899.9903.9908 9.8.99.997.99.996.9930.9934.9939.9943.9948.995 9.9.9956.996.9965.9969.9974.9978.9983.9987.999.9996 6