SOLUTION: Can I see the steps of how to solve: Log6X= 1/2 log6 9 + 1/3 log6 27

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
----------
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

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

To work with logarithms we need to be familiar with their properties and how to use them:
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.

RELATED QUESTIONS
Sara solved the following equation and knows she made a mistake. Explain to her where... (answered by Alan3354,josgarithmetic)
how do you solve log6x+ log6 (x+16)= 2... (answered by solver91311)
solve... (answered by solver91311)
how do i solve log6 3 + log6 8 - log6... (answered by Gogonati)
I need help with this question. solve the equation:... (answered by poliphob3.14)
1. 2^(5-3x)=1/16 how you do this and whats the answer 2. log6x+log6(x-3)=2 how you do... (answered by ewatrrr)
Hi I couldnt figure out how to do this problem... (answered by stanbon)
Solve Log6(x+3) + Log6(x-2) = 1 * 6 is the base for both logs (answered by user_dude2008)
Find the unknown in the equation using steps please:) log6 7 = log6(6^x+1 plus 6^x+2) (answered by stanbon,scott8148)