By Spiegel M.R., Stephens L.J.

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PROPERTIES OF LOGARITHMS The following are the more important properties of logarithms: 1. logb MN ¼ logb M þ logb N 2. logb M=N ¼ logb M À logb N 3. logb M P ¼ p logb M EXAMPLE 36. Write logb ðxy4 =z3 Þ as the sum or diﬀerence of logarithms of x, y, and z. xy4 ¼ logb xy4 À logb z3 property 2 z3 xy4 logb 3 ¼ logb x þ logb y4 À logb z3 property 1 z xy4 logb 3 ¼ logb x þ 4 logb y À 3 logb z property 3 z logb LOGARITHMIC EQUATIONS To solve logarithmic equations: 1. 2. 3. 4. 5. Isolate the logarithms on one side of the equation.

The continuous compounding produces slightly better results. PROPERTIES OF LOGARITHMS The following are the more important properties of logarithms: 1. logb MN ¼ logb M þ logb N 2. logb M=N ¼ logb M À logb N 3. logb M P ¼ p logb M EXAMPLE 36. Write logb ðxy4 =z3 Þ as the sum or diﬀerence of logarithms of x, y, and z. xy4 ¼ logb xy4 À logb z3 property 2 z3 xy4 logb 3 ¼ logb x þ logb y4 À logb z3 property 1 z xy4 logb 3 ¼ logb x þ 4 logb y À 3 logb z property 3 z logb LOGARITHMIC EQUATIONS To solve logarithmic equations: 1.

0. 45, and À3 in (a) increasing and (b) decreasing order of magnitude. 18), they increase from left to right. , solve each inequality for X): 3 À 2X 7 (a) 2X < 6 (c) 6 À 4X < À2 (e) À1 5 X À5 (b) 3X À 8 ! 4 (d ) À3 < <3 2 SOLUTION (a) Divide both sides by 2 to obtain X < 3. (b) Adding 8 to both sides, 3X ! 12; dividing both sides by 3, X ! 4. (c) Adding À6 to both sides, À4X < À8; dividing both sides by À4, X > 2. Note that, as in equations, we can transpose a term from one side of an inequality to the other simply by changing the sign of the term; from part (b), for example, 3X !