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Exercises for Logs
Converting log equations to exponential equations
Convert the following logarithmic equations into exponential equations, and solve for the
unknown if possible.
log 5 𝑥 = 𝑦
log 2 𝑥 = 12
ln 𝑦 = 4.918
log 𝑎 = −1.42
ln 𝑥 = ln 50
Converting exponential equations to log equations
Convert the following exponential equations into logarithmic equations, and solve for the
unknown if possible.
10𝑥 = 20
𝑒 𝑦 = 85
𝑎𝑐 = 𝑏
𝑒 −𝑦 = 0.452
2𝑥 = 128
Combining logs
(Combining logs is important for solving log equations.)
Express the following expressions using a single logarithm.
log 6 + log 𝑦
log(7) − log (10)
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3
log √𝑥 + 3log (𝑥 2 )
1
log 𝑎
3
− 2 log 6
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This document last updated: 7/5/2012
Student Academic Learning Services
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Solving log equations
Hint: The key here is to combine all the logs into one log, and then rearrange it into an
exponential equation before solving.
5 log 2 𝑥 = 20
log(8𝑥 2 ) − log(2𝑥) = 3
ln(2𝑥 + 3) − 2 ln 7 = 1.5
Solving exponential equations (with logs)
Hint: The key step is to take the log of both sides. It may help to rearrange it first though.
In all the solutions, I simplify fully in log form before writing the approximate answer as a
decimal number. You may not have to do that for class.
10𝑥 = 500
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𝑒 −2𝑥 − 1 = 4
2(5𝑥+2 ) = 300
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 7/5/2012