SOLUTION: [inside brackets = Base] (log[5](3x+10))-(3log[5](4))=2 Solve the equation. Thanks.

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Question 277571: [inside brackets = Base]
(log[5](3x+10))-(3log[5](4))=2
Solve the equation.
Thanks.
Found 2 solutions by jsmallt9, Alan3354:
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%285%2C+%283x%2B10%29%29+-+3log%285%2C+%284%29%29=2
We want the equation in the form:
log(expression) = other-expression
So somehow we need to combine the two logarithms into one. These two logarithms are not like terms so we cannot subtract them. But there is a property of logarithms, log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29, which can be used to combine two logarithms if all of the following are true:
  • there is a "-" between them
  • the bases of the logarithms are the same
  • the coefficients of the logarithms are 1's

Your logarithms meet the first two but not the last. So now our goal is to get rid of the 3 in front of the second log. And fortunately there is another property of logarithms, q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29, which can be used to move a coefficient of a logarithm into its argument as an exponent. Using this on your second log we get:
log%285%2C+%283x%2B10%29%29+-+log%285%2C+%284%5E3%29%29=2
which simplifies to:
log%285%2C+%283x%2B10%29%29+-+log%285%2C+%2864%29%29=2
These are still not like terms so we still cannot subtract them. But we can now use the other property to combine them:
log%285%2C+%283x%2B10%29%29%2F64%29%29=2
We now have the desired form. Once we have this form the next step is to rewrite the equation in exponential form:
%283x+%2B+10%29%2F64+=+5%5E2
which simplifies to:
%283x%2B10%29%2F64+=+25
Now the variable is out of the argument where we can "get at it". Solving this for x we start by multiplying both sides by 64 to get rid of the fraction:
3x + 10 = 1600
Subtracting 10 from each side:
3x = 1590
Dividing by 3 we get:
x = 530

When solving logarithmic equations, it is important (not just a good idea) to check your answers. Even if we've done everything correct so far, we need to make sure that each answer makes the argument of any logarithms positive.

Always use the original equation to check:
log%285%2C+%283x%2B10%29%29+-+3log%285%2C+%284%29%29=2
Checking x = 530:
log%285%2C+%283%28530%29%2B10%29%29+-+3log%285%2C+%284%29%29=2
which simplifies to
log%285%2C+%281600%29%29+-+3log%285%2C+%284%29%29=2
Both arguments are positive so it looks good. (You're welcome to finish the check on your own.)

The solution to your equation, then, is x = 530}}}

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
[inside brackets = Base]
(log[5](3x+10))-(3log[5](4))=2
Solve the equation.
-----------------------
log%285%2C3x%2B10%29+-+3log%285%2C4%29+=+2
log%285%2C3x%2B10%29+-+log%285%2C64%29+=+2
log%285%2C%283x%2B10%29%2F64%29+=+2
(3x+10)/64 = 25
3x+10 = 1600
3x = 1590
x = 530