# Solving for t, lost on progression to final step [calculus].

From an example question in my calc textbook:

Solve for $t$ in two steps, using a calculator at the final stage:

$$t\log 1.034 = \log P - \log 12.853$$

$$t = \frac{\log P}{\log 1.034} - \frac{\log 12.853}{\log 1.034}$$

$$t = 68.868 \log P - 76.375$$

Right, so I know that the $76.375$ comes from dividing $\dfrac{\log 12.853}{\log 1.034}$, but I can't for the life of me figure out what happened to get to $68.868 \log P$. Any help?

If the logarithms are common logarithms, i.e. to base $10$, you can see on a calculator that

$$\frac 1{\log 1.034}\approx 68.8679681017$$

Therefore we get

$$\frac {\log P}{\log 1.034}\approx 68.868\log P$$

Is the link between the equations now clear?