Why do we multiply by 10 when changing the subject of a formula with logarithms?

Question

I have an equation such as,

$$\ln(t) = -7.5 + 1.5\ln(d)$$

When needing to make $d$ the subject, online calculators multiply both sides by $10$.

Why do we multiply by $10$? I know that the base of $ln$ is 10 when it does not state a base.

Answer 1

This is to make those rational coefficients integers -- and is just the equivalent of clearing fractions, which is just to simplify the equation, so we now have

$$10\ln t=-75+15\ln d,$$ which simplifies further to $$2\ln t=-15+3\ln d.$$

PS. Your last sentence is wrong. The ugly notation $\ln$ is sometimes used to denote the logarithm when the base is a certain real number universally called $e.$ This may be given by the $\sum_{k=0}^{\infty}{1/k!}.$