# Logarithms with Variables

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Remember that

logb y = x

can be defined as an exponential expression like this:

y = bx

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

• ﻿log2 y = x
y = 2x
• log2 y = 3
y = 23
• log3 y = 4
y = 34
• log3 y = 7
y = 37
• log4 y = 3
y = 43
• log4 y = 5
y = 45

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

logb (n + 1) = x

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

(n + 1) = bx

• ﻿log2 (n + 1) = x
n + 1 = 2x
• log2 (n + 1) = 3
n + 1 = 23
• log3 (n + 2) = 4
n + 2 = 34
• log3 (n + 4) = 5
n + 4 = 35
• log4 (n + 6) = 3
n + 6 = 43
• log4 (n + 7) = 5
n + 7 = 45

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

• ﻿8 = 2x
log2 8 = x = 3
• 27 = 3x
log3 27 = x = 3
• 25 = 5x
log5 25 = x = 2
• 36 = 6x
log6 36 = x = 2
• 500 = e.05t
ln 500 = .05t
• 600 = 300e.07t
ln 2 = .07t
• 1500 = 500e.08t
ln 3 = .08t
• 2500 = 1000e.09t
ln 2.5 = .09t

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