# Solving logarithmic equation

I'm having trouble solving this equation. I know there is a solution as my graphics calculator can solve it, but I want to see the steps on how to get the answer.

The mathematical equation is:

$$\log_{10}n = 0.07n$$

• There is a solution, but it cannot be obtained by a finite number of algebraic manipulations. It can be expressed in terms of the Lambert-W function, which see; it can be obtained to any number of decimal places by numerical methods; it can't be expressed exactly in finite terms using the functions of high school math (or college math, actually). – Gerry Myerson Jul 23 '13 at 9:13
• If you plot the function $\log_{10}(x) - 0.07 x$ you see that there are two solutions, one near $x=1$ and one near $x=18$. You can find them with the Lambert-W functions or simply with root solving techniques (Newton, bisection etc). – gammatester Jul 23 '13 at 9:24
• excellent, thank you! – Ryan Jul 23 '13 at 9:49