# Deriving logarithmic identities

Wikipedia says:

The numerical value for logarithm to the base $10$ can be calculated with the following identity:

$$\mathrm{log_{10}}(x) = \frac{\mathrm{ln}(x)}{\mathrm{ln}(10)} \\ \text{or}$$

$$\mathrm{log_{10}}(x)= \frac{\mathrm{log_{2}}(x)}{\mathrm{log_{2}}(10)}$$

How are these identities derived? Can you provided a step by step to show how they are derived, I am a little confused about this and also about the notation.

Mainly, different ways of writing the same thing, such as:

$$\mathrm{log_{e}}(x)=\mathrm{ln}(x)$$ confuses me.

I would like to understand these identities more by seeing the step by step process as to how they were derived, this would also clarify the notational confusion. Thanks.

Suppose that $y = \log_{10}{x}$; then by definition, $10^y = x$.
Taking logarithms gives $\ln{x} = \ln{10^y} = y \ln{10}$ so that $$y = \frac{\ln{x}}{\ln{10}}$$