Logarithms: The Mystery Explained



(Created Tuesday, December 19, 2000. Edited Wednesday, December 20, 2000.)

(Last edited 14 December 2001.)

Created for LSCHS Mathletes web page (). Please do not place on other sites.

Logarithms: The Mystery Explained

As you may remember, inverse functions are two functions whose operations undo each other and whose composition is the function y=x (for example, the functions y=x3+4 and y=[pic].)

The exponential function is the function y=ax, where a>0 and a[pic]1. This function is useful for various reasons, but its inverse is not easily intelligible. That’s where logarithms come in.

The logarithm function is the inverse of the exponential function:

If y=ax, then x=loga y. a is termed the base of the logarithm.

For example, because 8=23, 3=log2 8. And, because the exponential and the logarithm are inverses, [pic]= loga ax = x.

The range of the exponential is all positive numbers; the domain is all Reals.

The domain of the logarithm function is all positive numbers; the range is all Reals.

Common Bases

The two most frequently used bases for logs are 10 and e.

• The log base 10 of a number, the common log, is usually written log x; the 10 is understood.

• Examples:

• log 10000=4

• log .001=-3

• log 1=0

• The log base e of a number, the natural log, is written ln x.

• e [pic] 2.71828. If you really care where this number comes from:

• [pic] (remember, 0!=1)

• And, for you calculus addicts, e is defined in these ways:

• [pic], and

• [pic].

The Laws of Logarithms

loga (xy)=loga x+loga y

loga ([pic])=loga x-loga y

loga (xn)=n loga x

logb x=[pic], where a and b are any valid base (The base change formula)

loga ax=x

[pic]=x

loga a=1

loga 1=0 for any valid base a

Logs on the Calculator

Often, problems will ask for the values of expressions like log 6 78. Using the base change formula,

log 6 78=[pic]=[pic]=[pic], where a is any valid base

So, entering “ln 78/ln 6” should yield the answer, as should entering “log 78/log 6”.

Practice problems

Without a calculator:

Given log 2=.3010, what is log 2 50?

Solve for x: 2log 6 5 = log 6 x + 2 (Keep in fractional form.)

Enjoy!

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