Step by step (show-your-work) example on how to solve a log problem algebraically

Question

It's been a long time since I have done calculus so if someone could please refresh my memory on the steps to solve the following problem algebraically that would be most appreciated. I am interested the step-by-step process of solving the problem algebraically without a calculator (if applicable), rather than just an answer (which is $n \approx 43.411$). The problem is as follows; solve for $n$: $$n \leq 8 \log n$$ Thanks in advance

Answer 1

These kind of problems usually need the http://en.wikipedia.org/wiki/Lambert_W_function, see the examples section. To get a solution near your $\approx 43.411,\;$ I guess the base of the logarithm is 2. Actually there are two solution for $n=8\log_2(n):$ $$n = \frac{-8}{\ln 2} W\Big(\frac{-8}{\ln 2}\Big) \approx 1.099997$$ or $$n = \frac{-8}{\ln 2}W_{-1}\Big(\frac{-8}{\ln 2}\Big) \approx 43.559260$$

If you do not want to use the Lambert W function, you can simply iterate the formula $n = 8 \log_2(n),\;$ using a starting value $n_0.\;$ Here are two iterations with $n_0=20, \quad n_0=40: $

n = 40.0
n = 8*log[2](n)  n = 42.57542476
n = 8*log[2](n)  n = 43.29559214
n = 8*log[2](n)  n = 43.48918599
n = 8*log[2](n)  n = 43.54067840
n = 8*log[2](n)  n = 43.55433585

n = 20.0
n = 8*log[2](n)  n = 34.57542476
n = 8*log[2](n)  n = 40.89340056
n = 8*log[2](n)  n = 42.83036906
n = 8*log[2](n)  n = 43.36449763
n = 8*log[2](n)  n = 43.50753993
n = 8*log[2](n)  n = 43.54554831

Answer 2

May be, you could consider the function $$f(x)=x-a \log_b(x)=x-\frac{a \log (x)}{\log (b)}$$ for which $$f'(x)=1-\frac{a}{x \log (b)}$$ $$f''(x)=\frac{a}{x^2 \log (b)}$$ The derivative cancels for $x_*=\frac{a}{\log (b)}$ and $$f(x_*)=\frac{a-a \log \left(\frac{a}{\log (b)}\right)}{\log (b)}$$ $$f''(x_*)=\frac{\log (b)}{a}$$ So, if $f(x_*)<0$ and $f''(x_*)>0$ as in the present case, two roots have to be and they are expressed in terms of Lambert function as gammatester answered.