- #1
Kjos
- 5
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Homework Statement
i.2(lgx)^2 - lgx = 0
and
ii. lg(2x-2)^2= 4lg(1-x)
and
iii. lgx-6 / lgx-4 = lgx.
I simply do not manage to solve these equations, and I would therefore be happy for all help. Thanks in advance.
Dickfore said:Hints:
ii) Notice that: lg (2x - 2) = lg[(-2)(1 - x)]. Now, if lg represents a principal value of the logarithm with a branch cut along the negative real axis, then:
[tex]
\mathrm{lg} \left[(-2)(1 - x) \right] = \mathrm{lg}(e) \cdot \ln{\left[(-2)(1-x) \right]} = \left\lbrace
\begin{array}{ll}
\mathrm{lg}(e) \cdot [\ln(2) + \ln(1-x)] = \mathrm{lg}(2) + \mathrm{lg}(1 - x) &, \ \mathrm{Arg}(1-x) \le 0 \\
\mathrm{lg}(e) \cdot [\ln(2) + \ln(1-x) - 2 \pi \, i] = \mathrm{lg}(2) + \mathrm{lg}(1 - x) - 2 \pi \, \mathrm{lg}(e) \, i &, \ \mathrm{Arg}(1-x) > 0
\end{array}\right.
[/tex]
Dickfore said:First of all, what do you mean by "lg"? Second, what is the domain in which you solve this equations? Is it complex numbers, or are you restricted to real numbers only?
Kjos said:Homework Statement
i.2(lgx)^2 - lgx = 0
and
ii. lg(2x-2)^2= 4lg(1-x)
and
iii. lgx-6 / lgx-4 = lgx.
I simply do not manage to solve these equations, and I would therefore be happy for all help. Thanks in advance.
Ray Vickson said:(ii) Is ambiguous. Do you mean
[lg(2x-2)]^2 = 4 lg(1-x), or do you mean
lg[(x-2)^2] = 4 lg(1-x)?
(iii) As written, (iii) is
(lg x) - (6/lg x) - 4 = lg x. Did you mean that? Or, did you mean
lg(x-1)/lg(x-4) = lg x? Or did you mean
[lg(x-1)/lg x] - 4 = lg x?
You need to use brackets to make things clear.
RGV
Kjos said:My mistake. [PLAIN]http://bildr.no/view/1297631[/QUOTE] [Broken]
So, what are the answers to the questions I asked? Also, someone asked you what "lg" represents, and you said "logarithm". But, there are three kinds of logarithms most commonly used : (i) logarithms to base 2; (ii) logarithms to base 10; and (ii) natural logarithms. Just saying "logarithm" does not really help us; you need to specify what type of logarithm you mean.
RGV
Ray Vickson said:So, what are the answers to the questions I asked? Also, someone asked you what "lg" represents, and you said "logarithm". But, there are three kinds of logarithms most commonly used : (i) logarithms to base 2; (ii) logarithms to base 10; and (ii) natural logarithms. Just saying "logarithm" does not really help us; you need to specify what type of logarithm you mean.
RGV
lendav_rott said:If the symbol is given as logx it is the default one, with the base of 10. In other cases you have the logarithm given as lognX - where n is the base and it is equal to log10X/log10n and finally you have the natural logarithm ln X which has a base of e - so ln X = logeX. Hope that clarifies it for you :)
A logarithm equation is a mathematical expression that uses logarithms, which are the inverse functions of exponential functions. It is used to solve for the unknown variable in an exponential equation.
Logarithm equations are commonly used in home settings for financial planning, such as calculating interest rates, investments, and mortgage payments. They can also be used to solve for time and growth rates in home improvement projects.
To solve a logarithm equation at home, you can use a calculator or follow a step-by-step process. First, isolate the logarithm term on one side of the equation. Then, convert the logarithm into an exponential form. Finally, solve for the unknown variable using basic algebraic principles.
Yes, some common mistakes when solving logarithm equations at home include forgetting to convert the logarithm into exponential form, incorrectly using the rules of logarithms, and not checking for extraneous solutions. It is important to double-check your work and use a calculator if needed.
Aside from financial planning and home improvement projects, logarithm equations can also be used to calculate pH levels in pools and hot tubs, measure decibels in home sound systems, and determine the half-life of household cleaning products.