<p>3.3.1 Expanding & Condensing Log Expressions: Applying the Log Laws By Dan I. Alex M. </p><p>Andrew V.</p><p>1) Log (6 * 11) 2) Log (a * b)^2</p><p>= Log 6 + Log 11 = 2 Log a + 2 Log b</p><p>3) Log (x * y * z^2) 4) Log 3 - Log 8</p><p>= Log x + Log y + 2 Log z = Log (⅜)</p><p>5) (2Log7)/3 6)2(log 2x - log y) - (log 3 + 2 log 5)</p><p>= Log 7^2 =Log (4x^2) / 75y^2</p><p>3.3.2 solving exponential equations 1) 200(1.05)^x=1000 2) 20(2.5)^x-400=600 x=32.98 x=4.26 3)160(0.5)^x=8 4) x^1.05=2 x=4.322 x=1.935</p><p>3.3.1 Log Laws By Dan I. Alex M. Andrew V.</p><p>1) Log of a product: Log(xy)= Log(x)+Log(y) 2) Log of a quotient: Log(x/y) = Log(x)-Log(y) 3) Power Law: Log(x^n) = (n)Log(x) These laws are shown for Logarithms base 10, but they are true for Logarithms with any base. The following proof is done for Logarithms base 10, but it also holds for any other base: Prove: Log(xy) = Log(x)+Log(y) Let Log x=M so that 10^M=x Definition of Log Let Log y = N, and 10^N=y Definition of Log (xy)= 10^M * 10^N = 10^M+N Rules of Exponents Log(xy)=Log(10^M+N) Take Log of both sides Log(10^M+N)=M+N Definition of Log Log(xy)=Log(x) + Log(y) Substitution</p><p>Expanding & Condensing Log Expressions: Applying the Log Laws 1) Log (6 * 11) 2) Log (a * b)^2</p><p>3) Log (x * y * z^2) 4) Log 3 - Log 8</p><p>5) (2Log7)/3 6)2(log 2x - log y) - (log 3 + 2 log 5)</p><p>3.3.2 How can I change the Base?</p><p>Solving Equations of the Form a^x=b. Given Equation a^x=b Take the log of both sides Log a^x =Log b Power Law for the Logs X Log a = Log b Divide both sides by Log a x= (log b)/(log a) Use a calculator to evaluate x=x</p><p>1) 200(1.05)^x=1000 2) 20((2.5)^x)-400=600</p><p>3)160(0.5)^x=8 4) x^1.05=2</p>