Equation with logarithmic and polynomial terms

In summary, the conversation is about an old exam problem from 1921 that involves solving an equation with both polynomial terms and logarithms. The conversation includes an attempt at simplifying the equation and a discussion about using logarithms with a base of 6. The conversation concludes with the realization that the equation cannot be solved using elementary functions and the possibility of using the Lambert W function or a logarithm table.
  • #1
NanakiXIII
392
0
This is not actually a homework question, but it seemed appropriate to put it here. In an old exam from 1921 I found the following problem. I never learned how to solve this type of thing and I haven't been able to figure it out, so: how does one solve this?

Homework Statement



Solve for [itex]x[/itex]:

[tex]\frac{(x-1)^2}{(x-1)-^6\log (x-1)} = 3 \times 6^{3\times^6\log 2 + 2\times^6\log 3}[/tex]

Homework Equations





The Attempt at a Solution



I went ahead and simplified this to

[tex]y^2 -72y + 72 ^6\log y=0[/tex]

where [itex]y=x-1[/itex], but, as I said, I never learned how to solve this type of equation involving both polynomial terms and logarithms and I don't know how to proceed.
 
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  • #2
NanakiXIII said:
This is not actually a homework question, but it seemed appropriate to put it here. In an old exam from 1921 I found the following problem. I never learned how to solve this type of thing and I haven't been able to figure it out, so: how does one solve this?

Homework Statement



Solve for [itex]x[/itex]:

[tex]\frac{(x-1)^2}{(x-1)-^6\log (x-1)} = 3 \times 6^{3\times^6\log 2 + 2\times^6\log 3}[/tex]
You have several instances of expressions such as 6log (something). I suspect that these really mean log6(something). IOW, the log expressions are log-base 6. Please clarify.
NanakiXIII said:

Homework Equations





The Attempt at a Solution



I went ahead and simplified this to

[tex]y^2 -72y + 72 ^6\log y=0[/tex]

where [itex]y=x-1[/itex], but, as I said, I never learned how to solve this type of equation involving both polynomial terms and logarithms and I don't know how to proceed.
 
  • #3
Yes, exactly. That's how we wrote log-base-6 in school.
 
  • #4
On the right, use the laws of logarithms: [tex]a log(x)= log(x^a)[/tex], log(x)+ log(y)= log(xy), and [tex]b^{log_b(x)}= x[/tex] to get [tex]3(6^{3log_6(2)+ 2log_6(3)})= 3(6^{log_6((2^3)(3^2)})= 3(8)(9)= 216[/tex] By the way, it is NOT a good idea to use "[itex]\times[/itex]" to indicate multiplication when you have [itex]x[/itex] as the unknown. Just use parentheses.
 
  • #5
Yes, it would appear I miscalculated, the 72's in my equation should be 216's. I'll edit my post. That said, though, I still don't know how to solve the equation.

Edit: Actually, it appears I can't edit the original post, so here's an erratum:

My simplification (last equation) should be

[tex]y^2 - 216 y - 216 \log_6 y = 0[/tex]

where [itex]y = x-1[/itex].
 
Last edited:
  • #6
Since that equation involves both powers of y and logarithm of y, the solution cannot be written in terms of "elementary" functions. It should be possible, by taking the exponential of both sides, to get it in the form [itex]ve^v= \text{constant}[/itex] and then solve it in terms of the Lambert W function.
 
  • #7
Well, that explains why I never learned to solve this. However, since this is an old high school exam, I would be surprised if there was not a simpler solution. It is probably possible to solve it using elementary functions and a logarithm table.
 

FAQ: Equation with logarithmic and polynomial terms

1. What is a logarithmic term in an equation?

A logarithmic term is a mathematical expression that contains a logarithm function, such as log(x) or ln(x). This function is used to represent the inverse relationship between a base number and its exponent. In an equation, a logarithmic term would typically be written as log(x) = a, where "a" is a constant.

2. How do you solve an equation with logarithmic and polynomial terms?

To solve an equation with logarithmic and polynomial terms, you can use algebraic manipulation and the properties of logarithms to isolate the logarithmic term and solve for the variable. If the equation also contains polynomial terms, you can use algebraic methods such as factoring or the quadratic formula to solve for the variable.

3. What are the key properties of logarithms that are important to know?

The key properties of logarithms are the product rule, quotient rule, and power rule. The product rule states that the logarithm of a product is equal to the sum of the logarithms of each individual factor. The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. The power rule states that the logarithm of a number raised to an exponent is equal to the product of the exponent and the logarithm of the base.

4. Can you have both logarithmic and polynomial terms in the same equation?

Yes, it is possible to have both logarithmic and polynomial terms in the same equation. In fact, many real-world problems involve equations with a combination of logarithmic and polynomial terms. For example, an equation modeling population growth or decay may contain both types of terms.

5. What are some applications of equations with logarithmic and polynomial terms?

Equations with logarithmic and polynomial terms are commonly used in fields such as science, engineering, finance, and economics. They can be used to model growth and decay, calculate interest rates, and solve various real-world problems. For example, the Richter scale for measuring earthquake magnitude uses logarithmic and polynomial terms.

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