Let's just say you have that base (call this base b) raised to some power p, equal to the value you wanted to take the logarithm of, V.

b^{p} = V

First let's just take the log of both sides with base b

\log_{b} b^{p} = \log_{b} V

Apply the Logarithm Power Rule

p \cdot \log_{b} b = \log_{b} V

The logarithm with base b of b is just

1

p = \log_{b} V

Now let's just take the logarithm of both sides of the first equation with a base of 10.

\log_{10} b^{p} = \log_{10} V

Apply the Logarithm Power Rule

p \cdot \log_{10} b = \log_{10} V

Now just divide

p = \frac{\log_{10} V}{\log_{10} b}

Replace p with the logarithm with base b of V

\log_{b} V = \frac{\log_{10} V}{\log_{10} b}

The key ideas here are knowing that if you apply something to both sides of an equation, they remain equal; And the Logarithm Power Rule.

Change of Base Formula