Log Equation Sample Questions

Here are some questions to go with the post on solving log equations. Answers and explanations are down at the bottom.

Find the exact value of x in each question:

2)   log5x – log52 = 3


2)   log2x – 2 = log23


3)   log64(x + 1) =  log64x + 1/2


4)   2log9(x – 3) = 4


5)   log8(x + 1) =  1/3 – log8x

Answers below, scroll down…
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1)   log5x – log52 = 3
     log5(x/2) = 3
     x/2 = 53 = 125
     x = 250

2)   log2x – 2 = log23
     log2x – log23 = 2
     log2(x/3) = 2
     x/3 = 22 = 4
     x = 12

3)   log64(x + 1) =  log64x + 1/2
     log64(x + 1) – log64x = 1/2
     log64((x + 1)/x) = 1/2
     (x + 1)/x = 641/2 = 8
     x + 1 = 8x
     7x = 1
     x = 1/7
NOTE: "Exact answer" means leave your answer as a fraction unless
it's an exact decimal. So 0.142857 would not be right.

4)   2log9(x – 3) = 4
     log9(x – 3) = 2
     (x – 3) = 92 = 81
     x = 84

ALTERNATIVE METHOD:
4)   2log9(x – 3) = 4
     log9(x – 3)2 = 4
     (x – 3)2 = 94
     (x – 3) = + or – 92 = +81 or –81
     x = 84 or x = –78
     x = 84 because you can't have the log of a negative number
     

5)   log8(x + 1) =  1/3 – log8x
     log8(x + 1) + log8x = 1/3
     log8((x + 1)x) =  1/3
     (x + 1)x = 81/3 = 2
     x2 + x = 2
     x2 + x - 2 = 0
     (x + 2)(x – 1) = 0
     x = –2 or x = 1
     x = 1 because you can't have the log of a negative number

Tags: logsquestions