How many ways are there of expressing how many 2’s we can multiply to make 8 2 x 2 x 2 = 8 - Count them three 2’s 23 = 8 - Note the exponent 3 log2 (8) = 3 - Express as a logarithm.

How many ways are there of expressing how many 2’s we can multiply to make 8 2 x 2 x 2 = 8 - Count them three 2’s 23 = 8 - Note the exponent 3 log2 (8) = 3 - Express as a logarithm. The number of 2s we need to multiply to get 8 is 3 therefore the logarithm is 3.

Working with Logs The base: the number we are multiplying (i.e. 2) Notice we are dealing with three numbers log2 (8) = 3 The base: the number we are multiplying (i.e. 2) How many times to use it in a multiplication (3 times, this is the logarithm) The number we want to get to (i.e. 8)

Working with Logs Example calculation log5 (25) = ___ This is asking us how many times we need to multiply the base (5) by itself (the logarithm) to find the value in the bracket.

Working with Logs Example calculation log5 (25) = 2 This is asking us how many times we need to multiply the base (5) by itself (the logarithm) to find the value in the bracket.

Find the Logarithm Try a few – NO CALCULATORS log5 (25) = ____

Find the Logarithm Try a few – NO CALCULATORS log5 (25) = 2

Logs and Exponents So a logarithm tells us how many times we need to multiply the base by itself to find a value. An exponent also does the same thing 23 = 8 log2(8) = 3

Consider Following… bK = c If; b = 2 k = 3 Then 23 = 8 Therefore c = 8

Consider the following… bK = c What if c and b are known but K is not. e.g. 2k = 16 Guessable while K remains small but much harder once things get large (or very small) e.g. 2K = 8,388,608

Consider the following… Since logarithms and exponents are inverse expressions we can take a log (logarithm) to find K (the exponent)…. Like so… 2k = 16 log2(16) = k i.e. How many times do we need to multiply 2 by itself to get 16… answer is 4.

Consider the following… But what about… 2K = 8,388,608 log2(8,388,608) = k How many times do we need to multiply 2 by itself to get 8,388,608… now we’ll need a calculator.

Consider the following… But what about… 2K = 8,388,608 log2(8,388,608) = k How many times do we need to multiply 2 by itself to get 8,388,608… now we’ll need a calculator.

Logs on the Calculator Because your calculator has been designed by an engineer the log button has a preset base. The log button is set to do log10, the most commonly used base. In fact so commonly used that when you see log(8) = X it means log10(8) = X therefore X is 0.90. Try these… Log10(10) = 1.00 Log10(200) = 2.30 Log10(1000) = 3.00 Log(100) = 2.00 Log(95421) = 4.98 Log(542) = 2.72

Logs on the Calculator There is another preset more often used by mathematicians which uses Euler's number as the base (2.71828). This number can be found by doing some complicated work with triangles and circles… BUT occurs “naturally”. We therefore call this a natural log. You will find it as a preset on your calculator as “ln”.

Logs on the Calculator log2(8,388,608) = ____ ln is the same as writing log 2.71828 Therefore these two expressions mean the same log 2.71828(8) = 2.08 ln(8) = 2.08 Try these… log(5987) = _____ log10(1120) = ____ ln(85) = _____ ln(380) = ____ ln(X) = 3.22 log2(8,388,608) = ____

Logs on the Calculator log2(8,388,608) = ____ ln is the same as writing log 2.71828 Therefore these two expressions mean the same log 2.71828(8) = 2.08 ln(8) = 2.08 Try these… log(5987) = 3.78 log10(1120) = 3.05 ln(85) = 4.44 ln(380) = 5.94 ln(X) = 3.22 log2(8,388,608) = ____

ln(x) = 3.22 When moving a log from one side of an equation to another we must use the inverse function (like x and ÷) For a log we use the exponent so… ln(x) = 3.22 Becomes x = e^3.22 (x = 25.03) You will find the e symbol as 2nd function ln.

ln(x) = 3.22 The same goes for a log where 10 is the base Log10(x) = 3 Becomes x = 10^3 You will find the 10x as 2nd function log.

log2(8,388,608) = ____ If you’ve been paying attention you will have noticed I STILL haven’t answered this equation. This requires you to use a log with 2 as the base – not a preset on your calculator therefore you will need to convert from log10 to log2. You can do so using this equation loga(x) = log x / log a If you are fancy you may have the ability to set your own logs on your calculator….

Logs on the Calculator Converting logs: loga(x) = log x / log a E.g. X = log8 / log2 X = 3

Logs on the Calculator … 23  Converting logs: loga(x) = log x / log a So finally what is log2(8,388,608) … 23 

Practice Page log2.71828 (4) = x log(30) = x ln(x) = 3.02 ln(30) = x log(8.2X108) = x log10(980) = x ln(42) = x

Practice Page log(30) = 1.48 log2.71828 (4) = 1.39 ln(30) = 3.40 log(8.2X108) = 8.91 log10(980) = 2.99 ln(42) = 3.74 log2.71828 (4) = 1.39 ln(20.49) = 3.02 log(3467.37) = 3.54 log6(1296) = 4 Log4.5(900) = 4.52