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Inverse Tricks

Do you remember what inverse functions do to each other?

THEY UNDO EACH OTHER!

So...

y = e^( x ) and y = Ln( x ) undo each other!
y = 10^( x ) and y = log( x ) undo each other!
y = 5^( x ) and y = log to the base 5( x ) undo each other!
Notice that the bases match on all of these.

So, in general,

y = a^( x ) and y = log to the base a( x ) undo each other!

Why am I putting exclamation points on all of these?  Because this is VERY exciting stuff (and I'm a geek).  Just wait until the next section
-- we'll be able to solve equations with this stuff!

Here's how you usually see these tricks written:

1 ) log to the base a( a^( x ) ) = x
 
2 ) a^( log to the base a( x ) ) = x

Here are some examples of rule 1 ) :

log to the base 5( 5^( 3 ) ) = 3

log( 10^( 8 ) ) = 8

 Ln( e^( 4 ) ) = 4

 

TRY IT:

log to the base 7( 7^( 2 ) ) = ___

log to the base 9( 9^( 6 ) ) = ___

 
Here are some examples of rule 2 ) :

4^( log to the base 4( 2 ) ) = 2

10^( log( 6 ) ) = 6

 e^( Ln( 3 ) ) = 3

 

TRY IT:

5^( log to the base 5( 8 ) ) = ___ 2^( log to the base 2( 9 ) ) = ___

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