- #1
songoku
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Homework Statement
Find x that satisfies:
log11 (x3 + x2 - 20x) = log5 (x3 + x2 - 20x)
Homework Equations
logarithm
The Attempt at a Solution
x3 + x2 - 20x = 1
x3 + x2 - 20x - 1 = 0
Then stuck...
Actually folks, this is correct:songoku said:Homework Statement
Find x that satisfies:
log11 (x3 + x2 - 20x) = log5 (x3 + x2 - 20x)
Homework Equations
logarithm
The Attempt at a Solution
x3 + x2 - 20x = 1
x3 + x2 - 20x - 1 = 0
Then stuck...
bossman27 said:I'm not sure what you're trying to do in your attempt at a solution...
Think of it this way:
[itex] log_{11}(x^{3}+x^{2}-20 x) = log_{5}(x^{3}+x^{2}-20x) [/itex]
means [itex] 11^{k} = 5^{k} = x^{3}+x^{2}-20x [/itex].
So what can "k" be for this: [itex] (11^{k} = 5^{k}) [/itex] to hold? I think you might have assumed it must be 1, but that is not correct.
songoku said:x3 + x2 - 20x - 1 = 0
Then stuck...
ehild said:You are not lucky with your book of problems. That equation has only complex roots. Check it at wolframalpha.com.
http://www.wolframalpha.com/input/?i=x^3+%2B+x^2+-+20x+-+1+%3D+0
ehild
songoku said:Hm...using my calculator, I got three real roots. They are:
x = - 4.9776
x = 4.02750
x = - 0.04988
And those numbers are really similar to the x-intercept of root plot (diagram) of wolframalpha. But I don't understand why there are also three alternatives complex solutions.
OK, the point is this equation can't be solved manually.
Thanks a lot for the help
The fact that the roots given by WolframAlpha are complex, is likely an artifact of whatever numerical method is being used there. The imaginary part of the roots listed there is extremely small relative to the real part, with the real part matching the roots as you have listed them and the imaginary part on the order of 10-15 .songoku said:Hm...using my calculator, I got three real roots. They are:
x = - 4.9776
x = 4.02750
x = - 0.04988
And those numbers are really similar to the x-intercept of root plot (diagram) of wolframalpha. But I don't understand why there are also three alternatives complex solutions.
OK, the point is this equation can't be solved manually.
Thanks a lot for the help
A logarithm equation with different base is an equation that involves a logarithm function with a base other than the typical base 10 or base e. The equation is written as logb(x) = y, where b is the base, x is the argument, and y is the result.
The main difference between a logarithm equation with different base and a standard logarithm equation (log10(x) or ln(x)) is the base used. In a standard logarithm equation, the base is either 10 or e, whereas in a logarithm equation with different base, the base can be any positive number other than 1.
To solve a logarithm equation with different base, you can use the change of base formula: logb(x) = log(x) / log(b). This formula allows you to convert the equation into a standard logarithm equation, which can then be solved using basic algebraic techniques.
Logarithm equations with different base have many applications in fields such as finance, computer science, and physics. They are commonly used in financial modeling, calculating exponential growth, and measuring the performance of algorithms. They are also used in physics to model radioactive decay and in chemistry to calculate pH levels.
The properties of logarithm equations with different base are similar to those of standard logarithm equations. Some of the key properties include the product rule, quotient rule, and power rule. These properties allow you to simplify and solve complex logarithm equations with ease.