91Ó°ÊÓ

Understanding the properties of logarithms is crucial when solving logarithmic equations. The properties allow us to simplify and manipulate logarithmic expressions to make them more manageable.

In this exercise, we use the property that states the difference of two logarithms can be expressed as a single logarithm. Specifically, the property is:

\[ \text{ln}(a) - \text{ln}(b) = \text{ln} \bigg( \frac{a}{b} \bigg) \]

This property enables us to combine the logarithmic terms into one, simplifying the equation and making it easier to solve.

So, for the equation \(\text{ln}(x+1) - \text{ln}(x-2) = 1\), we combine the logarithms as follows:

\[ \text{ln} \bigg( \frac{x+1}{x-2} \bigg) = 1 \]

Exponentiation is the inverse operation of taking a logarithm. When we have a logarithmic equation in the form \(\text{ln}(y) = b\), we can rewrite it as an exponential equation: \(y = e^b\), where \(e\) (approximately equal to 2.718) is the base of natural logarithms.

In the given exercise, we have:

\[ \text{ln} \bigg( \frac{x+1}{x-2} \bigg) = 1 \]

To solve for \(x\), we exponentiate both sides to remove the logarithm. This means we rewrite the equation in its exponential form:

\[ \frac{x+1}{x-2} = e^1 \]

Since \(e^1\) simplifies to \(e\), the equation becomes:

\[ \frac{x+1}{x-2} = e \]

Cross-multiplication is a method used to eliminate fractions by moving terms across an equation. This is particularly useful when dealing with ratios or proportions.

In our case, we have the equation:

\[ \frac{x+1}{x-2} = e \]

To eliminate the fraction, we cross-multiply by multiplying both sides by \(x-2\):

\[ x+1 = e(x-2) \]

This step simplifies our equation to a linear form, making it easier to solve for \(x\).

After simplifying the equation through cross-multiplication, the next step is solving for \(x\).

Starting from:

\[ x+1 = e(x-2) \]

We distribute \(e\) on the right-hand side:

\[ x + 1 = ex - 2e \]

Next, we collect all \(x\)-terms on one side and constants on the other side:

\[ x - ex = -1 - 2e \]

Factor out \(x\) on the left-hand side:

\[ x(1 - e) = -1 - 2e \]

Finally, solve for \(x\) by dividing both sides by \(1 - e\):

\[ x = \frac{-1 - 2e}{1 - e} \]

This is the solution to the original logarithmic equation.