Question 222578: Can I see the steps of how to solve:
Log6X= 1/2 log6 9 + 1/3 log6 27 Found 2 solutions by Alan3354, jsmallt9:Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website! Can I see the steps of how to solve:
Log6X= 1/2 log6 9 + 1/3 log6 27
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I assume those are logs base 6:
All base 6, tho the base doesn't matter if they're all the same.
log(x) = log(3) + log(3)
log(x) = log(9)
x = 9
This allows us to "move a coefficient into the argument as an exponent (and vice versa).
This allows us to combine the sum of to two logarithms into the log of the product of the arguments (or to split the log of a product into the sum of the logs).
This allows us to combine a difference of logarithms into the log of the quotient of the arguments (or take the log of a quotient and split it into a difference of the logs of the numerator and denominator).
To solve your equation we want to transform it so looks like:
log(x) = some simple expression.
Once we have it in that form we will then be able to solve for x. So we will use these properties to condense the right side of your equation into a simpler expression. Since the 2nd and 3rd properties require no coefficients, we'll start by using the 1st property to "move: them into the arguments as exponents:
Before continuing we'll simplify the arguments. Since 1/2 as an exponent means square root and since the square root of 9 is 3, And 1/3 as an exponent means cube root and since the cube root of 27 is 3, . So now our expression is:
Now we can use the 2nd property above to "add" these logs:
The simplest way to solve this is to relize that this equation says "the log (base 6) of x is the same as the log (base 6) of 9. And I hope it makes sense that if the log of x and the log of 9 are the same, then x must be the same as 9. IOW: x = 9.
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