Test Review Logarithms Exponentials Rational Exponents WECHS 10

Test Review Logarithms Exponentials Rational Exponents WECHS 10 th Grade Math December 16, 2010 Test December 17

Test Topics � Logarithms ◦ ◦ ◦ Convert between log and exponential form Evaluate logs – get numerical answer Expand logs using log properties Simplify logs using log properties Solve exponential equation using log � Rational exponents ◦ Convert between radical and exponential form ◦ Evaluate – get numerical answer � Exponential review – calculate growth or decay

Logarithm Basic & Exponential Form Swap between forms: LOG EXPONENTIAL

Properties of Logs Product Property � Logb(x·y) = Logbx + Logby Quotient Property � Logb(x ÷ y) = Logbx - Logby Power Property � Logbxy = y·Logbx Log of Base � Logbb = 1 Log of 1 � Logb 1=0

Evaluating Logs When you have the question “logba = x” (b and a will be numbers), ask “b to what power equals a? ” Ex: � Log 327 = ? 3 to what power equals 27? � Log 525 = ? 5 to what power equals 25? � Log. 01 = ? 10 to what power equals. 01? If you have a fractional base, flip the fraction over and make the answer negative. Log⅓ 9 = ? Log 39 = -x -x = 2 x = -2

Expanding Logs � Each factor should be split off into its own log, using the Product or Quotient Property. ◦ Log 6(2 xy) = Log 62 + Log 6 x + Log 6 y into separate logs before using the Power Property to remove exponents. � Split ◦ Log 54 x 2 = Log 54 + Log 5 x 2 = Log 54 + 2 Log 5 x � Except – if a power applies to more than one factor, do that first. ◦ Log(2 x)3 = 3 Log 2 x = 3(Log 2 + Logx)

Simplifying Logs � The opposite of expanding – combine everything into a single log if possible. ◦ Logx + Logy + Log 6 = Log 6 xy � Move factors outside the log up to be exponents before combining terms. ◦ 3 Logx + 2 logy = Logx 3 + Logy 2 = Logx 3 y 2

Solving Equations with Logs � If you have an equation with variable in the exponent, take log of both sides. � Ex: solve 3 x = 18

Rational Exponents �A rational exponent is a fraction in the exponent: ◦ Ex: � This x ⅓ , y ⅗, z ⅝ can be written in radical or exponent form:

Converting Radical ↔Exponents � Write in radical form: � Write in exponent form:

Evaluating Rational Exponents � Take root first if possible so you work with smaller numbers; then raise to a power. � You can raise to a power first if it is not possible to take the root.

Exponential Property Review �xm·xn = x(m+n) �. �(xm)n � . = xmn

Exponential Growth & Decay � Exponential functions show things that grow or decay/depreciate at a constant rate: �A is the initial amount (principle). � r is the growth rate (as a decimal) � If it is a growth problem, r is positive and the number in ( ) is greater than 1 � If it is a decay problem, r is negative, and the number in ( ) is less than 1.