# Solving Complex Logarithmic Equations

• duki
Therefore, the final expression can be simplified to ln\left(\frac{120}{8}*\frac{x}{x^1^/^3}*\frac{y}{y^3}\right) = ln\left(\frac{15yx^2^/^3}{y^2}\right). In summary, when simplifying ln expressions, you can use the properties of logarithms to combine and simplify terms. In this case, using the property ln(a)-ln(b)=ln(a/b) and the fact that x^p/x^q =x^(p-q), we can simplify the expression to ln\left(\frac{15yx^2^/^3}{y^2}\right).
duki

## Homework Statement

$$ln(6xy)-3ln(2y)-1/3 ln x + ln 20$$

The attempt at a solution

I can get to $$ln\left(\frac{120xy}{8y^3x^1^/^3}\right)$$

And then to

$$ln\left(\frac{120}{8}*\frac{x}{x^1^/^3}*\frac{y}{y^3}\right)$$

But then I am supposed to do something to get to $$ln\left(\frac{15yx^2^/^3}{y^2}\right)$$

What do I do? how does $$\frac{x}{x^1^/^3}$$ turn into $$x^2^/^3$$ and how does $$\frac{y}{y^3}$$ turn into $$y^2$$ ?

well x^p/x^q =x^(p-q)

same for y

o. simple. Thanks :)

Or
$$\frac{y}{y^3}= \frac{y}{y\cdot y\cdot y}= \frac{1}{y\cdot y}= \frac{1}{y^2}$$

## 1. How do you solve logarithmic equations?

To solve logarithmic equations, you must first rewrite the equation in exponential form. Then, use the properties of logarithms to simplify the equation and solve for the variable.

## 2. What are the properties of logarithms?

The properties of logarithms include the product, quotient, and power rules. 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 exponent times the logarithm of the number.

## 3. Can complex logarithmic equations have more than one solution?

Yes, complex logarithmic equations can have more than one solution. This is because logarithms are not one-to-one functions, meaning that different inputs can result in the same output. Therefore, it is important to check for extraneous solutions when solving complex logarithmic equations.

## 4. How do you check for extraneous solutions in logarithmic equations?

To check for extraneous solutions in logarithmic equations, you must substitute the solutions you have obtained back into the original equation. If the solution results in a negative or undefined value, it is considered extraneous and should be discarded.

## 5. What are some common mistakes to avoid when solving complex logarithmic equations?

Some common mistakes to avoid when solving complex logarithmic equations include forgetting to check for extraneous solutions, incorrectly applying logarithmic properties, and making errors in simplifying the equation. It is important to carefully review each step and check your work to avoid these mistakes.

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