Question 396430: How do I solve the following: log base 9 of x square + log base 3 of x = 5/6
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
Solving equations like this, where the variable is in the argument of a logarithm, usually starts by transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)
With the "non-log" term of 5/6 on the right side the second form, which is all logarithms, will be more difficult to achieve. So we will aim for the first form. The first form requires s single logarithm and your equation has two. Somehow we need to find a way to combine the two into one.
If the two logarithms were like terms then we could add them. But terms with logarithms are like terms only if both the bases and the arguments are the same. For example:  (Just like z + 3z = 4z!) Your two logarithms have difference bases, 9 and 3, and different arguments,  and x. So we cannot add them,
Another way to combine two logarithms into one is to use a property of logarithms. There are two such properties:
These properties require that the bases be the same and that the coefficients of the logs are 1's. Your two logarithms have bases that are different. However bases of logarithms can be changed using the base conversion formula:  . We can use this formula to convert the base 9 log into a base 3 log:
And since  ,  giving us:
We have made progress. The bases of the logarithms are equal. But we cannot use the property to combine the logs until we get rid of that 2 somehow. Since dividing by 2 is the same as multiplying by 1/2 we can rewrite the equation as:
This helps because there is yet a third property of logarithms,  , which allows us to move the coefficient of a logarithm into the argument as an exponent:
The rule for exponents when raising a power to a power is to multiply the exponents. And 2*(1/2) is 1!
We can now combine the logarithms. We can use the property or we can even add them now since the two terms are now like terms (the bases and arguments are now the same!). Adding them we get:
Dividing by 2 we get:
We have finally reached the first form! The next step with this form is to rewrite the equation in exponential form. In general  is equivalent to  . Using this pattern on your equation we get:
This is an exact expression for the solution to your equation. You may recall that fractional exponents can also be written in radical form:
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