# What is the inverse of  y=log(3x-1)?

Oct 30, 2015

$y = \frac{\log \left(x\right) + 1}{3}$

See the explanation

#### Explanation:

The objective is to get only $x$ on one side of the $=$ sign and everything else on the other. Once that is done you change the single $x$ to $y$ and all the $x ' s$ on the other side of the $=$ to $y$.

So first we need to 'extract' the $x$ from $\log \left(3 x - 1\right)$.

By the way, I assume you mean log to base 10.

Another way of writing the given equation is to write it as:

${10}^{3 x - 1} = y$

Taking logs of both sides

$\log \left({10}^{3 x - 1}\right) = \log \left(y\right)$

but $\log \left({10}^{3 x - 1}\right)$ may be written as $\left(3 x - 1\right) \times \log \left(10\right)$

and log to base 10 of 10 = 1
That is: ${\log}_{10} \left(10\right) = 1$

So no we have

$\left(3 x - 1\right) \times 1 = \log \left(y\right)$

$3 x = \log \left(y\right) + 1$

$x = \frac{\log \left(y\right) + 1}{3}$

Change the letters round

$y = \frac{\log \left(x\right) + 1}{3}$

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