My Math Forum Exponential and Logarithmic Equations help!

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 September 18th, 2013, 06:43 AM #1 Newbie   Joined: Jul 2013 Posts: 21 Thanks: 0 Exponential and Logarithmic Equations help! How do I solve these types of equations: $$\frac{10^{x}-10^{-x}}{2}=20$$, $$\frac{e^{x}+e^{-x}}{2}=15$$,$$\frac{1}{e^{x}-e^{-x}}=4$$, $$\frac{10^{x}-10^{-x}}{10^{x}+10^{-x}}=5$$, $$\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=3$$, etc? Step-by-step solutions would be helpful. Also, I've got this problem: In a town of 15,000 people, the spread of a rumor that the local transit company would go on strike was such that $t$ hours after the rumor, where experience over time has shown that $$f(t)=\frac{15000}{1+7499e^{-0.8t}}$$ a. How many people started the rumor? b. How many people heard the rumor after 5 hours? Now I know how to solve letter b by just substituting the $t$ for 5, but I would like to know how to solve letter a. Am I only going to substitute $t$ for 1 or what? And why? Thanks
 September 18th, 2013, 09:12 AM #2 Senior Member   Joined: Dec 2012 Posts: 372 Thanks: 2 Re: Exponential and Logarithmic Equations help! I will only solve your first problem, which will give you a cue to solve the rest. $10^x - 10^{-x}= 40 \Rightarrow 10^{2x} - 1 = 40(10^x)$ I simply multiplied thru by $10^x$. Now let $10^x= y$ and we observe that $y^2 - 40y -1= 0$. The rest is simple, you just solve the quadratic equation and substitute $x= \log_{10}y$. If $y$ is negative, then it doesn't yield any solution. For the number of people that started the rumor, you let $t= 0$ to get your answer.

### in a town of 15,000 people, the spread of a rumor

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