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You can put this solution on YOUR website!
start with:
\n" ); document.write( "xy - 10000y^2 = 9
\n" ); document.write( "since 10000 is the same as 10^4, this can be rewritten as:
\n" ); document.write( "xy - 10^4 * y^2 = 9
\n" ); document.write( "you are given that log(x) = 3 and you are given that log(y) = -2
\n" ); document.write( "using the basic definition of logs, you can derive the following from this:
\n" ); document.write( "log10(x) = 3 if and only if 10^3 = x
\n" ); document.write( "log10(y) = -2 if and only if 10^-2 = y
\n" ); document.write( "you can now replace x in your original equation with 10^3 and y in your original equation with 10^-2.
\n" ); document.write( "you will get:
\n" ); document.write( "xy - 10^4 * y^2 = 9 becomes:
\n" ); document.write( "10^3 * 10^-2 - 10^4 * (10^-2)^2 = 9
\n" ); document.write( "since 10^3 * 10^-2 = 10^1 and (10^-2)^2 = 10^(-4), you get:
\n" ); document.write( "10^1 - 10^4 * 10^-4 = 9
\n" ); document.write( "since 10^4 * 10^-4 = 10^0, you get:
\n" ); document.write( "10^1 - 10^0 = 9 which becomes:
\n" ); document.write( "10 - 1 = 9 which becomes:
\n" ); document.write( "9 = 9\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "solve for 5^(x+2) = 120 + 5^x
\n" ); document.write( "5^(x+2) is the same as 5^x * 5^2
\n" ); document.write( "your equation becomes:
\n" ); document.write( "5^x * 5^2 = 120 + 5^x
\n" ); document.write( "subtract 5^x from both sides of this equation to get:
\n" ); document.write( "5^x * 5^2 - 5^x = 120
\n" ); document.write( "factor out 5^x on the left side of the equation to get:
\n" ); document.write( "5^x * (5^2 - 1) = 120
\n" ); document.write( "simplify to get:
\n" ); document.write( "5^x * (24) = 120
\n" ); document.write( "divide both sides of this equation by 24 to get:
\n" ); document.write( "5^x = 120/24 which becomes 5^x = 5
\n" ); document.write( "if 5^x = 5, this means that x has to be equal to 1 because 5^1 is equal to 5.
\n" ); document.write( "you could also solve this by taking the log of both sides of the equation to get log10(5^x) = log10(5) which becomes:
\n" ); document.write( "x * log10(5) = log10(5) which becomes:
\n" ); document.write( "x = 1 after you divide both sides of the equation by log10(5).\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "solve for log3(x) = log9(6x+7)
\n" ); document.write( "log3(x) = y if and only if 3^y = x
\n" ); document.write( "log9(6x+7) = y if and only if 9^y = 6x+7
\n" ); document.write( "9^y is the same as (3^2)^y which is the same as 3^(2y) which is the same as (3^y)^2
\n" ); document.write( "you get:
\n" ); document.write( "3^y = x and (3^y)^2 = 6x+7
\n" ); document.write( "if you square both sides of 3^y = x, you get (3^y)^2 = x^2
\n" ); document.write( "now you have:
\n" ); document.write( "(3^y)^2 = x^2 and (3^y)^2 = 6x + 7
\n" ); document.write( "this means that x^2 = 6x + 7
\n" ); document.write( "subtract 6x + 7 from both sides of this equation and you get:
\n" ); document.write( "x^2 - 6x - 7 = 0
\n" ); document.write( "factor this equation to get:
\n" ); document.write( "(x-7) * (x+1) = 0
\n" ); document.write( "solve for x to get x = 7 and x = -1
\n" ); document.write( "those should be your solutions.
\n" ); document.write( "confirm by replacing x in your original equation with 7 and then with -1 to see if the original equations are true.
\n" ); document.write( "when x = 7, the original equation of log3(x) = log9(6x+7) becomes:
\n" ); document.write( "log3(7) = log9(6*7+7) which becomes:
\n" ); document.write( "log3(7) = log9(49)
\n" ); document.write( "in order to solve these, you can convert them to exponential form as we did above, or you can use the log base conversion formula to get:
\n" ); document.write( "log10(7)/log10(3) = log10(49)/log10(9) and use your calculator to get:
\n" ); document.write( "1.771243749 = 1.771243749, confirming the solution of x = 7 is good.
\n" ); document.write( "you can do the same with x = -1.
\n" ); document.write( "i'll confirm that one using the exponential form rather than the log base conversion formula.
\n" ); document.write( "your original equation is log3(x) = log9(6x+7)
\n" ); document.write( "when x = -1, you get:
\n" ); document.write( "log3(-1) = log9(1)
\n" ); document.write( "you can stop right there, because you can't take the log of a negative number if you are looking for a real solution.
\n" ); document.write( "x = -1 is not a real solution to the original equation.
\n" ); document.write( "the only real solution is x = 7.\r
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