Question 813675: This is my problem in logarithm
log x + log x^2 + log x^3 + log x^4 = 1 + log 0.2 + log 0.03^2 + log 0.004^3
thank's
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
A general procedure for solving these kinds of equations:- Use algebra and/or properties of logarithms to transform the equation into one of the following forms:
- log(expression) = other-expression
- log(expression) = log(other-expression) (Note: The bases of the two logs must match.)
- Eliminate the logarithms:
- If the equation is in the first form, "log(expression) = other-expression", rewrite the equation in exponential form.
- If the equation is in the second form, "log(expression) = log(other-expression)", set the arguments equal.
- Now that the logs are gone, solve the equation (using techniques which are appropriate for the type of equation it is).
- Check your solution. This is not optional! A check must be made to see if the bases and arguments of all logs are valid. Any "solution" which make any base or an argument invalid must be rejected! (Note: Valid bases are positive but not 1 and valid arguments are positive.)
Let's try this on your equation. First we decide which form we think will be easiest to achieve. With the "non-log" term of 1 (on the right side), it would seem that the second, "all-log" form will be harder to reach. So we will aim for the first form.
Stage 1: Transform
To reach this form, all we need to do is find a way to combine all the logs into a single logarithm. We will start by combining all the logs on each side into single logarithms. For this we will use the  property:
Simplifying...
Now we will get the logs on the same side. Subtracting the log on the right we get:
And now we can use another property of logarithms,  , to combine the remaining logs:
We have now reached the first form.
Stage 2: Eliminate the logs.
With the first form we just rewrite the equation in exponential form. In general  is equivalent to  . Using this pattern, and the fact that the base of "log" is 10, we get:
which simplifies to:
Srage 3: Solve
Our equation is now an exponential equation. But with only one x term it will not be hard to solve. Multiplying each side by 0.00000001152 we get:
Now we find the 10th root of each side (remembering both the positive and negative roots):
 or 
Stage 4: Check
Use the original equation to check:
Checking x = :
This can be done by inspection. This value of x is positive. So all the powers of x on the left side will be positive if x is positive. So all the bases arguments are valid. So this value checks out!
Checking :
With x being negative, even powers of x will be positive but odd powers of x will be negative. This makes the 1st and 3rd arguments negative. This is invalid. So we must reject this "solution".
So the only solution to is x = .
P.S. Technically, the solution should be rationalized. (A decimal is just a disguised fraction and radicals are not supposed to have fractions in them.) So the rest of this is rationalizing the solution:
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