Help!
how do i do these questions.
Determine the invers of these:
a)y=3log10(x)
b)y=2log10(3x)
Thanks
You original function was $\displaystyle f(x)= 3 log_{10}(x)$. The basic definition of "inverse functions" requires that $\displaystyle f^{-1}(f(x))= x$ and $\displaystyle f(f^{-1}(x))= x$. $\displaystyle 3 log_{10}(10^{3x})= 3(3x)= 9x,$ not x. Similarly $\displaystyle 10^{3 (3 log_{10}(x)}= 10^{9log_{10}(x)}= 10^{log_{10}(x^9)}= x^9$, not x.
A standard method of finding the inverse of a function given as y= f(x) is
1) Swap x and y (that's the key point).
2) Solve the equation for y.
3) If there is a unique solution for y as a function of x, that is the inverse.
Here, $\displaystyle y= 3 log_{10}(x)$. Swapping x and y, $\displaystyle x= 3log_{10}(y)$. To solve for y, first, divide both sides by 3: $\displaystyle \frac{x}{3}= log_{10}(x)$. Now take 10 to the power of both sides: $\displaystyle 10^{\frac{x}{3}}= 10^{log_{10}(y)}= y$. Since $\displaystyle y= 10^{\frac{x}{10}}$, that is the inverse function.