4
votes
3answers
227 views

How to solve $e^x = 2$

I know that $\ln(x)$ is the inverse of the exponential function $a^x$. So I thought that $$ e^x=2 \Leftrightarrow x = \ln(2) $$ but my calculator says $x = \ln(2) + 2 i \pi n$, where $N \in ...
4
votes
7answers
229 views

Solving $x^{\log(x)}=\frac{x^3}{100}$

How do I find the solution to: $$x^{\log(x)}=\frac{x^3}{100}$$ So I multiplied 100 both sides getting: $$100x^{\log(x)}=x^3$$ Now what should I do?
2
votes
1answer
54 views

How to compute a product of logarithms?

I've been reading through Stewart's Calculus textbook, and came across the following problem fairly early on - What is $$\prod_{i = 2}^{31} \log_i (i + 1)\;?$$ I did some searching, and found ...
4
votes
2answers
99 views

How to isolate x in this equation?

Suppose we have this equation: $y = 1 + x + \lceil \log_2(x)\rceil$ where $x$ is an integer > $0$. How can we get $x$ as a function of $y$ (basically isolate $x$)? I don't understand how to handle ...
8
votes
5answers
226 views

Solving an equation with a logarithm in the exponent

I try to solve the following equation: $$ (N+1)^{\log_N{125}} = 216 $$ I know the answer is 5 here but how could I rewrite the equations so I can solve it? I tried to take the log of both sides but ...
4
votes
3answers
441 views

Solving $xe^{-x}+2e^{-x}=0$

While I was studying my maths book, I came across this equation: $$ xe^{-x}+2e^{-x}=0 $$ I tried to solve it in different ways, but each time I break up some rule. My best try was this: Let's ...
13
votes
4answers
440 views

How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?

How do I integrate this guy? I've been stuck on this for hours.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
0
votes
2answers
41 views

How to evaluate this formula

How can I evaluate $1/e^{\ln(x)}$? I really don't have experience on this and appreciate if you can explain it to me. Thanks.
0
votes
2answers
2k views

Compound interest - how to solve this with logarithms & geometric series?

I could use some help with the following: Jacques is saving for a new car which will cost 29000 dollars. He saves by putting 400 dollars a month into a savings account which gives 0.1% interest ...