# Logarithm rearrangement

1. Mar 21, 2017

### OnlinePhysicsTutor

1. The problem statement, all variables and given/known data
2 - log10 3x = log10(x/12)

2. Relevant equations
logab=b log a
log(a/b)= log a - log b

3. The attempt at a solution
2 + log10 12= log10 x - x log10 3
Start seems simple but cannot see where to go from here, taking exponentials doesn't seem to help. Not sure what the next steps could be.

2. Mar 21, 2017

### Buffu

log is base 10 or it is natural log ?

3. Mar 21, 2017

### OnlinePhysicsTutor

I assume it is meant to be base 10, so I have edited post to include the base.

4. Mar 21, 2017

### Buffu

After rearranging a bit I can't see how this has a nice solution.

I get $5^2 *2^4*3 = x3^x$.

5. Mar 21, 2017

### OnlinePhysicsTutor

I'm happy with that as well, thanks.

6. Mar 21, 2017

### Staff: Mentor

Your equation is equivalent to $100 = \frac x {12} \cdot 3^x$. Because the variable occurs both as an exponent and as a multiplier, there are not any simple analytic ways to solve this equation. However, you can get good approximations by numeric means, simply by substituting value for x on the right side, and comparing the result with 100 on the left side. Using a spreadsheet I see that there is a solution near x = 50.16. The actual solution is slightly smaller than this.

Edit: As Ray points out, my number here is incorrect. I was using an incorrect formula in my spreadsheet, using log(3^x) instead of 3^x.

Last edited: Mar 21, 2017
7. Mar 21, 2017

### Ray Vickson

For If $f(x) = (x/12) 3^x,$ we have$f(50) = (50/3) e^{50} \doteq 0.299 \times 10^{25}$, so the solution of $f(x) = 100$ must certainly be quite a bit less than 50. Maple gets $x \doteq 4.990.$

8. Mar 21, 2017

### Buffu

what is $\doteq$ ?

9. Mar 21, 2017

### Ray Vickson

$\doteq$ means "approximately equal to", sometimes also written as $\approx$. I avoid using "=" in such cases just so the reader will understand that the answer is not exactly 4.990. For example, a better approximation is obtained by using 60 digits of precision, giving
$x \doteq 4.99043541467729841484302401855197675632523233638262678465047$ Even that is not exact.

10. Mar 21, 2017

### Buffu

As a additional exercise, Can we prove that there is no nice real solution for equation, Also can we know the nature of the solution ?

11. Mar 21, 2017

### Staff: Mentor

You are correct. Somehow I mistakenly had log(3^x) in my spreadsheet formula, not log(3^x) as it should have been.