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January 28th, 2012, 12:58 AM   #1
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Hard logarithm question

Hello!

Attached is a very hard logarithm question that I couldn't answer in my math test. If someone could help me answer it, I would very much appreciate it

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 January 28th, 2012, 01:13 AM #2 Senior Member   Joined: Jul 2011 Posts: 227 Thanks: 0 Re: Hard logarithm question You are given: $2(5^{x+1})=1+\frac{3}{5^x}$ Your first step to make a common denominator for the RHS is a good start: $2(5^{x+1})=\frac{5^x+3}{5^x}$ Now use some logarithm rules: $2(5^{x+1})5^x=5^x+3$ $\Leftrightarrow 2(5^{2x+1})=5^x+3$ $\Leftrightarrow 2(5^{2x}\cdot 5)=5^x+3$ $\Leftrightarrow 10\cdot 5^{2x}-5^x-3=0$ To continue make the substitution $5^x=t$ which will give you a quadratic equation in $t$
 January 28th, 2012, 01:20 AM #3 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Hard logarithm question We are given to solve: $2$$5^{x+1}$$=1+\frac{3}{5^x}$ $10\cdot5^x=1+\frac{3}{5^x}$ Multiply through by $5^x$ as we know $5^x\ne0$ for all real x: $10\cdot5^{2x}=5^x+3$ Arrange as quadratic on $5^x$: $10\cdot$$5^x$$^2-5^x-3=0$ Factor: $$$5\cdot5^x-3$$$$2\cdot5^x+1$$=0$ Assuming x is real, we must discard the negative root, and are left with: $5^x=\frac{3}{5}$ Convert from exponential to logarithmic form: $x=\log_5$$\frac{3}{5}$$=\log_5(3)-\log_5(5)=-1+\log_5(3)$
 January 28th, 2012, 02:23 AM #4 Newbie   Joined: Jan 2012 Posts: 3 Thanks: 0 Re: Hard logarithm question Thank you! I'm really impressed with this forum, the answers are clear and succinct
 January 22nd, 2017, 01:45 AM #5 Newbie   Joined: Jan 2017 From: Somewhere Posts: 1 Thanks: 0 Answer x = log[base 5](3) - 1 Last edited by Wizzy0630; January 22nd, 2017 at 01:47 AM.
 May 31st, 2017, 02:37 AM #6 Newbie   Joined: May 2017 From: China Posts: 1 Thanks: 0 Answer Hey, I know I may be too late to answer you now. You do not need to use logarithm for this question. Set 5^x as y, and rewrite the equation as 2*5*y = 1+3/y. Then continue the equation and factorize the quadratic you get. Keep going until you get your answer. Thanks

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