Logarithms and triangles help

In summary, Mr. Pythagoras said that "the sqaw on the hippopotamus" is c²=a²+b²...but then how do I apply that to the given statement. Perhaps if you give me a hint, I will manage to finish off the question.?The Attempt at a SolutionAssuming that a=2 and c=b+2, the equation can be rewritten as c²=a²+b²+c+b². This equation can be solved for b using the Quadratic Formula, and the result is b=2.
  • #1
L²Cc
149
0

Homework Statement



Define a, b, and c as the sides of a right triangle where c is the hypotenuse, and a > 1 and c > b+1

show that
[tex]log_{c+b} a + log_{c-b} a = 2(log_{c+b} a)(log_{c-b} a)[/tex]


2. Governing equations



The Attempt at a Solution


Should I assume that a=2 and c=b+2?!
 
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  • #2
L²Cc said:

Should I assume that a=2 and c=b+2?!


What did Mr. Pythagoras say about "the sqaw on the hippopotamus"?
 
  • #3
c²=a²+b²...but then how do I apply that to the given statement. Perhaps if you give me a hint, I will manage to finish off the question.
 
  • #4
?
 
  • #5
I haven't got time to look at it now, but maybe it would be useful, as mentioned above, to use a^2 = c^2 - b^2 = (c - b) (c + b), and the fact that [tex]\log_{a}x^n = n\log_{a}x[/tex] somehow.
 
  • #6
OK, here's the strategy I used to get a handle on this.

First off, it's a totaly weird looking equation. I can't think what practical use it would be. The things in there are "a", "c+b" and "c-b" It's also got logs to two different bases in the same equation.

I tried putting in the numbers for some rightangled triangles like 3,4,5 and 5,12,13. That didn't help much.

What else are you given? some weird stuff, a > 1 and c > b+1 or c-b > 1. Hm... maybe that's just saying all the "interesting" things are positive numbers so the logs are well defined. I don't know what else to do with it yet, so forget about it till later...

What else do we know? Well, a^2 + b^2 = c^2 is the only equation you have. So try and work forwards from a^2 + b^2 = c^2, and backwards from what you are trying to prove, and see if you can meet up in the middle.

From a^2 + b^2 = c^2 we want some stuff with c+b, c-b, and some logs in it. OK...
a^2 = c^2 - b^2 = (c+b)(c-b)
2 ln a = ln (c+b) + ln(c-b)

... that looks promising but all the logs are to the same base (e). So what happens if you work backwards from the answer and transform all the logs into natural logs? The standard formula is log(base p)q = ln(q)/ln(p).
 

What is the definition of a logarithm?

A logarithm is the power to which a base number must be raised to produce a given number. It is represented as logb(x) = y, where b is the base, x is the number, and y is the power.

How are logarithms used in triangles?

In triangles, logarithms are used to solve for unknown angles or side lengths. This is done by using the properties of logarithms and applying them to the trigonometric functions sine, cosine, and tangent.

What are the properties of logarithms?

The properties of logarithms include the product property, quotient property, power property, and the change of base property. These properties allow for easier manipulation and solving of logarithmic equations.

How are logarithms related to exponential functions?

Logarithms and exponential functions are inverse operations of each other. This means that if a number is raised to a power, the logarithm of that number will give the exponent. Similarly, if a logarithm is taken of a number, the power to which the base must be raised to produce that number.

What are some real-world applications of logarithms and triangles?

Logarithms and triangles have numerous applications in fields such as physics, engineering, and finance. They are used to calculate earthquake magnitudes, calculate pH levels in chemistry, and predict population growth. In engineering, they are used to measure signal strength and in finance, they are used in compound interest calculations.

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