# 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

## 1 Answer

Not all equations can be solved algebraically. This equation can not be solved step by step to find the answer, you will have to use limits to get to the answer (which is kind of what your calculater does).