log summary & equations

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


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:


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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27/04/12

log summary & equations

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