Change of Base Formula for Logarithms

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CHANGE OF BASE FORMULA FOR LOGARITHMS

Jump right to the exercises!

Before doing this exercise, you may want to review basic properties of logarithms:
Introduction to Logarithms
Properties of Logarithms

Recall that a logarithm is an exponent:
for example, log₂ 8 (log base two of eight) is the power that 2 must be raised to, to get 8 .
In this case, the numbers work out nicely: log₂ 8 = 3 , since 2³ = 8 .
But what if, say, you need to know log₂ 9 ?
You know it will be a little more than 3 , but suppose you need a six decimal place approximation?

Most calculators only have two built-in logarithms:

the natural logarithm (log base e ), denoted by ln
the common logarithm (log base 10), denoted by log

You can rummage around your calculator menus looking for logarithms to bases other than e or 10 , but you're not likely to find them.
What's a person to do?

The GOOD NEWS is that it is very easy to rename a logarithm with one base, as a logarithm with a different base.
All that is needed is the Change of Base Formula for Logarithms, which is the subject of this section.
Here's a preview of coming attractions:

changing to natural logs:   log2 9=ln   9ln   2  ≅calc   3.169925

changing to common logs:   log2 9=log   9log   2   ≅calc   3.169925

indeed, you can change to any allowable base:   log2 9=log 79 log72    (but this isn't a useful name for calculator computation)
You probably already see the pattern from these three examples.
Here's the precise statement:

Change of Base Formula for Logarithms
Let a and b be positive numbers that are not equal to 1 , and let x>0 .
Then, logb x=log a x loga b In words:
You can change from any base b to any base a ;
the "adjustment" is that you must divide by the log to the new base (a) of the old base (b).

Derivation of the Change of Base Formula for Logarithms

The following equations are equivalent:

y=logb x	Give a name, y , to the left-hand side of the Change of Base formula.
by =x	Write the equivalent exponential form of the equation.
loga   by =loga   x	Apply the function loga to both sides of the equation. (For more advanced readers: equivalence comes from the fact that loga is a one-to-one function.)
y  log a  b= log a  x	Use a property of logs to bring the y down.
y=log a x loga b	Divide both sides by loga   b . Compare with the first step!

You can use GeoGebra to explore the Change of Base formula for logarithms, by clicking here:

GeoGebra Worksheet: Change of Base Formula for Logarithms
(Please be patient. It may take a few minutes for GeoGebra to load. It's worth the wait!)

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.