# How do you solve log 5 = (x+1) log 4?

Dec 31, 2015

This would be same as solving a multi-step linear equation only the coefficients and constants are logarithmic numbers. Step by step working is shown below.

#### Explanation:

$\log \left(5\right) = \left(x + 1\right) \log \left(4\right)$

Let us distribute $\left(x + 1\right) \log \left(4\right)$

$\log \left(5\right) = x \log \left(4\right) + \log \left(4\right)$

What we see here is not different from any multi-step linear equation .
Say if you had no log and the equation was $5 = 4 x + 4$ you would have found it easy!

We shall use the same trick to solve it, but with the log present.

$\log \left(5\right) = x \log \left(4\right) + \log \left(4\right)$
First subtract $\log \left(4\right)$ on both the sides, this is done to isolate the $x$

$\log \left(5\right) - \log \left(4\right) = x \log \left(4\right)$

$\log \left(\frac{5}{4}\right) = x \log \left(4\right)$ Note : $\log \left(P\right) - \log \left(Q\right) = \log \left(\frac{P}{Q}\right)$

Now we divide both sides by $\log \left(4\right)$ to isolate $x$

$\log \frac{\frac{5}{4}}{\log} \left(4\right) = x$

The exact solution is $x = \log \frac{\frac{5}{4}}{\log} \left(4\right)$

If you want an decimal approximation then you can use a calculator to find it.