Can Excel be Used to Solve Logarithmic Equations with Different Bases?

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In summary: I believe all bob meant is 2x+1 for most numbers is really really close to 2x. Same idea for x+3 is close to x. In summary, BobG believes that most numbers are very close to 2x, so the change of base rule is not of much help. He also suggests using the laws of logarithms and using a numerical approximation to solve the equation.
  • #1
harimakenji
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Homework Statement


9 log3(2x+1) + 4 log2(x+3) = 85. Find x!


Homework Equations


logarithm


The Attempt at a Solution


I have tried few things but useless. Do not know what to do with different bases. I also tried to change the RHS: 85 = 5 x 17 but nothing worked.

Thank you very much
 
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  • #2


Do you know about the change of base rule?
 
  • #3


jgens said:
Do you know about the change of base rule?

I am not really sure what you meant. If this what you meant log3(2x+1) = log2(2x+1) / log23, I have tried that and do not know how to continue

Thank you very much
 
  • #4


Sorry! The change of base rule doesn't seem to be of much help here (at least, I couldn't make it work). WolframAlpha didn't give a closed form solution, just a numerical approximation, so this leads me to think that it probably doesn't have a nice and tidy answer.
 
  • #5


use logbx = logax / logab to get them all in the same base.

After algebraic simplification use the ratio of log properties.
 
  • #6


harimakenji said:
I am not really sure what you meant. If this what you meant log3(2x+1) = log2(2x+1) / log23, I have tried that and do not know how to continue

Thank you very much
Yes, that's right. And that changes the equation to
[tex]\frac{9}{log_2(3)}log_2(2x+1)+ 4log_2(x+ 3)= 85[/tex]

Now, as JonF suggested, use the laws of logarithms.
[tex]\frac{9}{log_2(3)}log_2(2x+1)= log_2((2x+1)^{9/log_2(3)}[/tex]
and
[tex]4 log_2(x+3)= log_2((x+3)^4)[/tex]
so we have
[tex]log_2((2x+1)^{9/log_2(3)}(x+3)^4}= 85[/tex]
and you can take 2 to the power of each side to get
[tex](2x+1)^{9/log_2(3)}(x+3)^4= 2^{85}[/tex]

Now, of course, that's still a pretty nasty equation! As jgens said, you probably will need to solve this numerically.
 
  • #7


HallsofIvy said:
Yes, that's right. And that changes the equation to
[tex]\frac{9}{log_2(3)}log_2(2x+1)+ 4log_2(x+ 3)= 85[/tex]

Now, as JonF suggested, use the laws of logarithms.
[tex]\frac{9}{log_2(3)}log_2(2x+1)= log_2((2x+1)^{9/log_2(3)}[/tex]
and
[tex]4 log_2(x+3)= log_2((x+3)^4)[/tex]
so we have
[tex]log_2((2x+1)^{9/log_2(3)}(x+3)^4}= 85[/tex]
and you can take 2 to the power of each side to get
[tex](2x+1)^{9/log_2(3)}(x+3)^4= 2^{85}[/tex]

Now, of course, that's still a pretty nasty equation! As jgens said, you probably will need to solve this numerically.

I got more or less same equation. Can you explain to me how to do it numerically? I do not think I need to do it that far but I am just curious.

Thank you very much
 
  • #8


Through incredibly awful calculus or punching it in a calculator. If some horrible teacher forced me to find a numerical approximation of x, I would first bring everything to the RHS, then call the LHS f(x). And do some numerical approximation of zeros method like Newtons or graph it very carefully and eyeball it.
 
Last edited:
  • #9


Or approximate it, assuming x was close to some power of 2, then brute force it with excel. Pull the 2 up into the power on the first term and...

That would give you approximately:

x^{6.678}*x^4=2^{85}
x^{10.678}=2^{85}
x=2^8


x is a little more than 256, which at least gives you a decent starting point if you're using Newton's method or a decent starting point if you decide to just brute force it. You can get to 291 (actually, closer to 292) very quickly just by brute forcing it, which is already more significant digits than you started with.
 
Last edited:
  • #10


JonF said:
Through incredibly awful calculus or punching it in a calculator. If some horrible teacher forced me to find a numerical approximation of x, I would first bring everything to the RHS, then call the LHS f(x). And do some numerical approximation of zeros method like Newtons or graph it very carefully and eyeball it.

BobG said:
Or approximate it, assuming x was close to some power of 2, then brute force it with excel. Pull the 2 up into the power on the first term and...

That would give you approximately:

x^{6.678}*x^4=2^{85}
x^{10.678}=2^{85}
x=2^8


x is a little more than 256, which at least gives you a decent starting point if you're using Newton's method or a decent starting point if you decide to just brute force it. You can get to 291 (actually, closer to 292) very quickly just by brute forcing it, which is already more significant digits than you started with.

I don't fully grab what BobG posted but I'll study the post more later. Until now, the teacher hasn't told the class how to solve the problem. Because it must be solved numerically, I guess I'll just wait the solution from the teacher.

Thank you for the help. I really appreciate it
 
  • #11


I believe all bob meant is 2x+1 for most numbers is really really close to 2x. Same idea for x+3 is close to x.

Let's say x is 50 (it's not, but let's pretend) then treating the 2x + 1 like it was just 2x would get you get you 100 instead of 101 for the 2x+1. I.e. it will be fairly close and make your numbers simpler to compute, so it will give you an approximation
 
  • #12


Oh it's like that. I understand now. How to brute force it with excel?
 

1. What is a logarithm?

A logarithm is a mathematical function that represents the relationship between a given number and its exponent. In simpler terms, it tells us how many times a certain number needs to be multiplied by itself to get another number.

2. What is a base in logarithms?

The base in logarithms is the number that is being raised to a power. For example, in the expression log28, 2 is the base and 8 is the number being raised to a power.

3. How do I solve a logarithmic problem with a different base?

To solve a logarithmic problem with a different base, you can use the change of base formula which states that logba = logca / logcb, where a is the number being raised to a power, b is the base, and c is the new base you want to use.

4. What is the common base used in logarithms?

In mathematics, the most commonly used base in logarithms is 10. However, in computer science and engineering, the base 2 and base e (Euler's number) are often used.

5. What is the purpose of using different bases in logarithms?

Using different bases in logarithms can help in simplifying complex mathematical expressions and calculations. It also allows for easier comparisons between numbers and can be useful in solving equations involving exponential functions.

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