Explain how you can transform the grap of f(x)= logx to produce g(x)=log(10nx), for any n>0.
log(10 n x) = log 10 + log n + log x
To transform the graph of f(x) = log(x) into g(x) = log(10nx), we'll go through a step-by-step process.
Step 1: Reflection in the y-axis
To reflect the graph of f(x) = log(x) in the y-axis, replace x with -x.
This gives us h(x) = log(-x).
Step 2: Horizontal stretch
To horizontally stretch the graph, replace x with a constant n times x.
This gives us p(x) = log(-nx).
Step 3: Horizontal translation
To horizontally translate the graph, replace x with x + a constant value.
This gives us q(x) = log(-nx - b), where b is a constant.
Step 4: Vertical translation
To vertically translate the graph, replace y with y + a constant value.
This gives us r(x) = log(-nx - b) + c, where c is a constant.
Step 5: Vertical stretch
To vertically stretch the graph, replace y with a constant n times y.
This gives us g(x) = n(log(-nx - b) + c).
Finally, we simplify the expression for g(x):
g(x) = log(10nx) = log(10n) + log(x) = n(log(-nx - b) + c)
Therefore, to transform the graph of f(x) = log(x) into g(x) = log(10nx), we need to reflect it in the y-axis, horizontally stretch it by a factor of n, horizontally translate it by some value, vertically translate it by another value, and finally vertically stretch it by a factor of n.