Logarithms
A Code for Exponents

When we look an any number raised to a power, that is any number with an exponent, we can think of the exponent as a logarithm or a log as represented by the log icon. Thus when

2³ = 8 can be expressed as log₂ 8 = 3

or we can say that the 'log with base 2 of 8' = 3.

For a power of 10 then,

10³ = 1,000 can be expressed as 3 = log₁₀ 1,000 = 3.

Evaluating Logs

We can use the Code to evaluate any log as in the two examples given in the picture above.

If x = log₂9 , then we can use the code to write

2^x = 9

From this we can easily see that x must = 3 and thus

log₂ 9 = 3.

If x = log₁₀ 1,000 , then we can use the code to write

10^x = 1,000

From this we can easily see that x must = 3 and thus

log₁₀ 1,000 = 3.

Examples

The picture above gives three examples of how to use the Code to solve log problems.

If we want to find the log₂ 32 we can express this in the Code as

2^x = 32

From this we can see that x must be 5 and so

log₂ 32 = 5

If we want to find the log₃ 27 we can express this in the Code as

3^x = 27

From this it is easy to see that x must be 3 and thus

log₃ 27 = 3

If we want to find the log₁₀ 1,000 we can express this in the Code as

10^x = 1,000

From this we see that x must be 3 and that

log₁₀ 1,000 = 3

Common Logs

There are two very common bases used for logs in mathematics. The first and most common base is 10.

If 10² = 100 then 2 is called the 'Log of 100 with base 10' and expressed as log₁₀ 100

The base of 10 is so common that

log₁₀ 100

is usually written simply as

log 100

and is referred to as the

Common Log of 100.

Natural Logs

The second most common base for logs is the unique, irrational number e = 2.718281828459... The number e is very important in calculus and higher mathematics and has a very unique property. The number e is the only number where the slope of y = e raised to any power (y = e^x))is equal to itself. That is the slope of the function y = e^x at any point = e^x.

An example logarithm with base e is

log_e 4

which is referred to as the

'log with base e of 4'

or more commonly the

'Natural Log of 4'

Thus the

'Natural Log of 4' = log_e = ln 4.

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