1. ## Logs- finding inverses

Help!

how do i do these questions.

Determine the invers of these:
a)y=3log10(x)
b)y=2log10(3x)

Thanks

2. Originally Posted by smmmc
Help!

how do i do these questions.

Determine the invers of these:
a)y=3log10(x)
b)y=2log10(3x)

Thanks
Hi

$\log_{10} x = \frac{y}{3} \rightarrow x = 10^{\frac{y}{3}}$

thanks

4. Originally Posted by smmmc

thanks
You original function was $f(x)= 3 log_{10}(x)$. The basic definition of "inverse functions" requires that $f^{-1}(f(x))= x$ and $f(f^{-1}(x))= x$. $3 log_{10}(10^{3x})= 3(3x)= 9x,$ not x. Similarly $10^{3 (3 log_{10}(x)}= 10^{9log_{10}(x)}= 10^{log_{10}(x^9)}= x^9$, 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, $y= 3 log_{10}(x)$. Swapping x and y, $x= 3log_{10}(y)$. To solve for y, first, divide both sides by 3: $\frac{x}{3}= log_{10}(x)$. Now take 10 to the power of both sides: $10^{\frac{x}{3}}= 10^{log_{10}(y)}= y$. Since $y= 10^{\frac{x}{10}}$, that is the inverse function.