Simplifying Logarithmic Ratios: How Do You Do It?

In summary, to simplify Ln (2x/y) / Ln (x/y) = m/n, you can cross-multiply and apply the properties of logarithms (ln(ab) = ln(a) + ln(b)) to get 1+ (ln 2)/Ln (X/Y) = m/n.
  • #1
casanova2528
52
0
how the heck do you simplify this ?

(Ln (2x/y) / Ln (x/y)) = m/n

HELP ME!
 
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  • #2
Start by cross-multiplying, then apply what you know about the log of an exponential expression.
 
  • #3
casanova2528 said:
how the heck do you simplify this ?

(Ln (2x/y) / Ln (x/y)) = m/n

HELP ME!

I am just going to elaborate a lill bit what marcusl already suggested.

You probbably know that

[tex] log\frac{x}{y}=log(x)-log(y)[/tex]

Also

[tex] log(ab)=log(a)+log(b)[/tex]

just apply these properties, and yu'll be fine.
 
  • #4
Ln (2x/y) = Ln 2x - Ln y

Ln (x/y) = Ln x - Ln y

[Ln (2x/y) / Ln (x/y)] = (Ln 2x - Ln y) / (Ln x - Ln y)

what do I do now?
 
  • #5
Well that wasn't how I interpreted the original hint. After cross-multiplying as already said, to [tex] n \log_e \left( \frac{2x}{y} \right) = m \log_e \left( \frac{x}{y} \right) [/tex], I would have applied the exponential function to both sides and simplified.
 
  • #6
Hell yeah.Gib Z is so right, my bad!
 
  • #7
Gib Z said:
Well that wasn't how I interpreted the original hint. After cross-multiplying as already said, to [tex] n \log_e \left( \frac{2x}{y} \right) = m \log_e \left( \frac{x}{y} \right) [/tex], I would have applied the exponential function to both sides and simplified.

that's not where I want to go.

basically, this natural log ratio reduces down to

1+ (ln 2)/Ln (X/Y)


how do you get here?
 
  • #8
casanova2528, you started with an equation (note the equals sign), so I assume you meant to write : 1+ (ln 2)/Ln (X/Y) = m/n.

To get this you should use the property of logs that ln(2x/y) = ln(2) + ln(x/y). You should find it pretty easy from there.
 
  • #9
uart said:
casanova2528, you started with an equation (note the equals sign), so I assume you meant to write : 1+ (ln 2)/Ln (X/Y) = m/n.

To get this you should use the property of logs that ln(2x/y) = ln(2) + ln(x/y). You should find it pretty easy from there.

thanks! Those darn properties!
 

1. What is a natural log ratio?

A natural log ratio is a mathematical expression that represents the logarithm of a ratio between two numbers. It is typically written as ln(a/b), where a and b are the two numbers in the ratio. The natural logarithm is the logarithm with base e, where e is the mathematical constant approximately equal to 2.71828.

2. How do you simplify a natural log ratio?

To simplify a natural log ratio, you can use the properties of logarithms to rewrite the expression in a simpler form. For example, the natural logarithm of a quotient can be written as the difference of the natural logarithms of the individual numbers: ln(a/b) = ln(a) - ln(b). You can also use the power rule, product rule, and quotient rule to further simplify the expression.

3. What is the purpose of simplifying a natural log ratio?

Simplifying a natural log ratio can make the expression easier to work with in calculations and can help in solving equations and inequalities involving logarithms. It can also make the solution more clear and understandable.

4. Are there any restrictions when simplifying a natural log ratio?

Yes, there are some restrictions when simplifying a natural log ratio. The numbers a and b must both be positive, and the value of a must be greater than the value of b. Additionally, the expression cannot be simplified if the ratio a/b contains variables or if a or b is equal to 1.

5. Can you simplify a natural log ratio with a negative number?

No, you cannot simplify a natural log ratio with a negative number. The natural logarithm function is only defined for positive numbers, so the expression ln(a/b) is not defined if a or b is negative. However, you can take the natural logarithm of the absolute value of the numbers and then use the properties of logarithms to simplify the expression.

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