SOLUTION: log base5 (3x+15)- log base 5 of 3x=1

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Question 115701: log base5 (3x+15)- log base 5 of 3x=1
Answer by solver91311(24713) About Me  (Show Source):
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log%285%2C3x%2B15%29-log%285%2C3x%29=1

You need to use three of the rules for logarithms for this problem.

1) log%28b%2Cx%5Ea%29=a%2Alog%28b%2Cx%29, and
2) log%28b%2Cf%28x%29%29%2Blog%28b%2Cg%28x%29%29=log%28b%2Cf%28x%29%2Ag%28x%29%29
3) log%28b%2Cx%29=y is equivalent to b%5Ey=x

Using the first rule, you can write:

log%285%2C3x%2B15%29%2Blog%285%2C3x%5E%28-1%29%29=1, notice that the minus sign in the original equation became an exponent of -1.

log%285%2C3x%2B15%29%2Blog%285%2C%281%2F3x%29%29=1 (using the rule x%5E%28-1%29=1%2Fx)

Now applying rule 2) we can write:
log%285%2C%283x%2B15%29%2A%281%2F3x%29%29=1

Now, using rule 3) we can write:
5%5E1=%283x%2B15%29%2A%281%2F3x%29

Simplify and solve for x
15x=3x%2B15
12x=15
4x=5
x=5%2F4. Done.