Learning Intention
Recap logarithmic expressions, rules & equationsSuccess criteria: you will be able toExpress log statements in exponential formApply log rulesSolve log equations
Back to Basics
Image source: www.purplemath.comA log is just the inverse of an exponential!
y = bx is equivalent to logb(y) = x(means the exact same thing as)
The Log Switcheroo
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Write the following exponential expressions in log form:63 = 21645 = 1024Write the following logarithmic expressions in exponential form:log2(8) = 3log5(25) = 2log64(4) = 1/3
log6(216) = 3log4(1024) = 523 = 852 = 25641/3 = 4
SIMPLE (base 10) EXAMPLES
NumberExponential ExpressionLogarithm
10001033
1001022
101011
11000
1/10 = 0.110-1-1
1/100 = 0.0110-2-2
1/1000 = 0.00110-3-3
Some Things To Remember
b1 = b , sologb(b) = 1, for any base b b0 = 1 , so logb(1) = 0logb(a) is undefined if a is negative logb(0) is undefinedlogb(bn) = n
Calculations with Logs
Because logarithms are exponents, mathematical operations involving them follow the same rules as those for exponents: 1) Multiplication inside the log can be turned into addition outside the log, and vice versa.logb(mn) = logb(m) + logb(n)2) Division inside the log can be turned into subtraction outside the log, and vice versa.logb(m/n) = logb(m) logb(n)3) An exponent on everything inside a log can be moved out front as a multiplier, and vice versa.logb(mn) = n logb(m)
Log Rule Practice
Expand log3(2x)log3(2x) = log3(2) + log3(x)Expand log4( 16/x )log4( 16/x ) = log4(16) log4(x)log4(16) = 2 so log4( 16/x ) = 2 log4(x)Expand log5(x3) log5(x3) = 3 log5(x) = 3log5(x)
Change of Base
eg Evaluate log3(6)log3(6) = log(6)= 1.63092975... log(3)
Solving a Log Equation
Step 1: Write as one log on one sideStep 2: Use the definition of logarithms to write in exponential form (or vice versa)Step 3: Solve for xeg log5(x+2) = 353 = x + 2125 = x + 2x = 123
A Trickier Example
Write as one log on one side
Use the definition of logarithms to write in exponential form
Solve for x
log(x + 21) + log(x) = 2log [(x+21)x] = 2
x2+21x = 102
x2+21x 100 = 0
(x+25)(x-4) = 0x = 4, -25BUT we CANNOT take the log of a negative number, so we will have to throw out x = -25 as one of our solutions
Remember that when there is no base written on a log, that means it is log base 10
Exam Question
Write as one log on one side
Use the definition of logarithms to write in exponential form (or vice versa)
Solve for x
H(t) = 3 + (1.24)tWhen does H = 7?7 = 3 + (1.24)t4 = (1.24)tlog (4) = log (1.24t)log (4) = t x log (1.24)t = log (4)= 6.44 years log(1.24)
Try the Following:
Write as one log on one side
Use the definition of logarithms to write in exponential form
Solve for x
log(2x 4) = 3log2(x) + log2 (x 6) = 4log4(x + 4) log4(x 1) = 2
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