document.write( "Question 619389: sketch y= -log5(-x) by first describing the transformation performed on the equation y=log5(x)
\n" ); document.write( "find the eqution of the inverse function
\n" ); document.write( "sketch theinverse fuction on the same axes \n" ); document.write( "
Algebra.Com's Answer #389614 by Theo(13342)\"\" \"About 
You can put this solution on YOUR website!
original equation is:
\n" ); document.write( "y = -log(5,-x)
\n" ); document.write( "the value of x would have to be negative in order to make log(-x) to be positive.
\n" ); document.write( "the graph of this equation would be:
\n" ); document.write( "\"graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C-log%285%2C-x%29%29\"
\n" ); document.write( "the inverse function of:
\n" ); document.write( "y = -log(5,-x) is found as follows:
\n" ); document.write( "multiply both sides of the equation by -1 to get:
\n" ); document.write( "-y = log(5,-x)
\n" ); document.write( "this equation is true if and only if:
\n" ); document.write( "5^(-y) = -x
\n" ); document.write( "multiply both sides of this equation by -1 to get:
\n" ); document.write( "-5^(-y) = x which is the same as:
\n" ); document.write( "x = -5^(-y)
\n" ); document.write( "replace x with y and y with x and you have the inverse equation of:
\n" ); document.write( "y = -5^(-x)
\n" ); document.write( "graph this equation to get:
\n" ); document.write( "\"graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C-5%5E%28-x%29%29\"
\n" ); document.write( "if these 2 equations are inverse equations of each other, then they will appear to be reflections about the line y = x.
\n" ); document.write( "graph all 3 equations together to get:
\n" ); document.write( "\"graph%28600%2C600%2C-5%2C5%2C-5%2C5%2Cx%2C-log%285%2C-x%29%2C-5%5E%28-x%29%29\"
\n" ); document.write( "they look like reflections about the line y = x.
\n" ); document.write( "we can test by taking one of the values and seeing if:
\n" ); document.write( "the coordinate pair (x,y) from the logarithmic equation is equal to:
\n" ); document.write( "the coordinate pair of (y,x) from the exponential equation.
\n" ); document.write( "we'll try x = -2
\n" ); document.write( "in the logarithmic equation, when x = -2, y = -.4306766
\n" ); document.write( "in the exponential equation, when x = -.4306766, y = -2
\n" ); document.write( "you have:
\n" ); document.write( "(x,y) from the logarithmic equation = (y,x) from the exponential equation.
\n" ); document.write( "this confirms they are inverse functions of each other.
\n" ); document.write( "the inverse equation undoes what the original equation does.
\n" ); document.write( "let f(x) = -log(5,-x)
\n" ); document.write( "let g(x) = -5^(-x)
\n" ); document.write( "if they are inverse equations, then g(f(x)) = x
\n" ); document.write( "we start with x = -2
\n" ); document.write( "we get f(x) = -.4306766
\n" ); document.write( "we then get g(f(x)) = g(-.4306766) which is equal to -2 which is equal to x.
\n" ); document.write( "since g(x) is the inverse function of f(x), then g(x) can be shown as f^(-1)(x) which means that g(x) is the inverse function of f(x).
\n" ); document.write( " \n" ); document.write( "
\n" );