# Logarithms with Variables

Remember that

log_{b}*y* = *x*

can be defined as an exponential expression like this:

*y* = *b ^{x}*

Many exercises expect you to convert from logarithmic to exponential expression as in these flash cards:

- log
_{2}*y*=*x*;*y*= 2^{x} - log
_{2}*y*= 3 ;*y*= 2^{3} - log
_{3}*y*= 4 ;*y*= 3^{4} - log
_{3}*y*= 7 ;*y*= 3^{7} - log
_{4}*y*= 3 ;*y*= 4^{3} - log
_{4}*y*= 5 ;*y*= 4^{5}

Many exercises introduce a variable into the *y* value like this:

log* _{b}* (

*n*+ 1) =

*x*

You can easily solve for *n* by converting to an exponential expression like this:

(*n* + 1) = *b ^{x}*

- log
_{2}(*n*+ 1) =*x*;*n*+ 1 = 2^{x} - log
_{2}(*n*+ 1) = 3 ;*n*+ 1 = 2^{3} - log
_{3}(*n*+ 2) = 4 ;*n*+ 2 = 3^{4} - log
_{3}(*n*+ 4) = 5 ;*n*+ 4 = 3^{5} - log
_{4}(*n*+ 6) = 3 ;*n*+ 6 = 4^{3} - log
_{4}(*n*+ 7) = 5 ;*n*+ 7 = 4^{5}

And if you are really **BOLD** you will use logarithms to solve for exponents:

- 8 = 2
^{x}; log_{2}8 =*x*= 3 - 27 = 3
^{x}; log_{3}27 =*x*= 3 - 25 = 5
^{x}; log_{5}25 =*x*= 2 - 36 = 6
^{x}; log_{6}36 =*x*= 2 - 500 =
*e*^{.05t}; ln 500 = .05*t* - 600 = 300
*e*^{.07t}; ln 2 = .07*t* - 1500 = 500
*e*^{.08t}; ln 3 = .08*t* - 2500 = 1000
*e*^{.09t}; ln 2.5 = .09*t*

Be sure to contact Fred if you would like further explanations of these flash cards.

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