Question 693278: How to solve the following equation log[9](5x-3)- log[3]2√x =0. Please help
Found 2 solutions by stanbon, jsmallt9:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! solve the following equation log[9](5x-3)- log[3]2√x = 0
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log9(5x-3) - [log9[2sqrt(x)]]/log9(3) = 0
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log9(5x-3) - [log9(2sqrt(x)]/(1/2) = 0
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log9(5x-3) - log9(4x) = 0
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log[(5x-3)/4x = 0
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(5x-3)/4x = 1
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5x-3 = 4x
x = 3
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Cheers,
Stan H.
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Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website! 
Solving equations like this usually starts with transforming the equation into one of the following forms:
log(expression) = number
or
log(expression) = log(other-expression)
If we can find a way to combine the two logarithms into one, then we would have the first form. There are two ways to combine log terms:- Adding (or subtracting) them. This requires that the logarithms have the same bases and the same arguments.
- Use a property of logarithms:
 when the terms has a "+" between them when the terms has a "-" between them
Both of these properties require that the logarithms have the same base and coefficients of 1.
Since both of these methods require logarithms of the same base and since our logs have different bases, we start by changing the base of one or both logs so that they are the same. To change bases we use the Change of Base formula:  . We need to change the base 9 log into an expression of base 3 logs or vice versa or change both bases into the same third number. Since 9 is well-known power of 2 (and vice versa for that matter) we will not need to change both bases. We can just change base 9 into base 3. (I'll show changing base 3 into base 9 at the end.) Using the change of base formula to change the base 9 log:
Since   :
To get rid of the fraction I'll multiply each side by 2:
Now that the bases are equal we can start to combine the terms. The arguments are different so they are not like terms. So we cannot subtract them from each other.
The properties require coefficients of 1 and the second log has a coefficient of 2. So we cannot use the properties, yet. Fortunately there is another property of logarithms,  , which allows use to move a coefficient into the argument as its exponent. Using this property we can move the coefficient of 2:
which simplifies to:
The logs now meet the requirements for the properties that combine terms. We'll use the second one because its logs, like ours, have a "-" between them:
We finally have the first form.
The next step with the first form is to rewrite the equation in exponential form. In general  is equivalent to  . Using this pattern on our equation we get:
which simplifies to:
Now that the variable is out of the logs we can solve. Multiplying by 4x:
Subtracting 4x:
Adding 3:
Last of all we check. This is not optional when solving logarithmic equations. You must at least ensure that all argument remain valid (i.e. positive) for each solution. Any "solution" that makes an argument invalid (zero or negative) must be rejected.
Use the original equation to check:
Checking x = 3:
We can already see that both arguments will be positive when x = 3. So this solution passes the check. (If x = 3 failed this check we would reject it and, since it was the only "solution" we found, it would mean that there were no solutions to the equation.)
P.S. Changing the base 3 log into base 9:
Since 3 is the square root of 9 and since an exponent of 1/2 means square root, the denominator is equal to 1/2:
which simplifies to:
This rest is very similar to what we did above and we end up witht he same solution.
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