# Defining the Range of Variables in Logarithmic and Radical Expressions

• feely
In summary, the equation ln(2+y) = 5ln(3 - x) - 2 sqrt x has a range of values for x and y in which both sides of the equation are defined. To find these values, one must determine the range of values for which ln(2+y) and sqrt x are defined. This can be done by considering the domain of the natural logarithm and square root functions, respectively. Once the range of values for these expressions is determined, one can solve for the range of values for x and y that satisfy the given equation.
feely

## Homework Statement

Real variables x and y are related by the equation

ln(2+y) = 5ln(3 - x) - 2 sqrt x ...(sorry, I haven't as yet got the hang on LaTeX)

Determine the range of values of x and y for which the expressions on each side of this equation are defined.

## The Attempt at a Solution

I haven't really been able to make an attempt at a solution. I think I have to take the exp of each side, but I am not sure excatly what way I go about this, so if someone could give me some advice, that would be fantastic.

Sean

HINTS:

(1) For what values of $\gamma$ is $\ln\gamma$ defined?

(2) For what values of $\eta$ is $\sqrt{\eta}$ defined?

## 1. What are logs and real variables?

Logs and real variables are mathematical concepts used to represent and manipulate quantities that vary continuously. Logs, short for logarithms, are the inverse of exponential functions and are used to represent the exponent needed to produce a given value. Real variables refer to variables that can take on any real number value, as opposed to discrete variables that can only take on specific values.

## 2. How are logs and real variables used in science?

Logs and real variables are used in various scientific fields such as physics, chemistry, and biology. They are often used to model natural phenomena that involve continuous change, such as growth, decay, and diffusion. They are also used in data analysis and mathematical modeling to make predictions and draw conclusions.

## 3. What is the relationship between logs and exponents?

Logs and exponents are inverse functions of each other. This means that if a number is raised to a certain power (exponent), the logarithm of that number will give the exponent needed to produce the number. For example, if 2^3 = 8, then log base 2 of 8 is equal to 3.

## 4. How do you solve equations involving logs and real variables?

To solve equations involving logs and real variables, you can use the properties of logarithms, such as the product, quotient, and power properties. You can also use the fact that the logarithm of a number with a base is equal to the exponent needed to produce that number. It is important to pay attention to the domain and range of the logarithmic function to ensure all solutions are valid.

## 5. Are there any real-life applications of logs and real variables?

Yes, logs and real variables have many real-life applications. They are used in finance to calculate compound interest and in population growth models. They are also used in signal processing to analyze sound and radio waves. In addition, they are used in computer science for data compression and encryption algorithms.

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