# 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

```

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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
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
```