Question

Write each expression as a single logarithm. \(\log \left(x^{2}+3 x+2\right)-2 \log (x+1)\)

Short answer

\[\log\left(\frac{x + 2}{x + 1}\right)\]

Step-by-step solution

Apply Logarithm Power Rule

Utilize the power rule of logarithms, which states that \(a \log(b) = \log(b^a)\). Apply this rule to the second term. The expression becomes: \[ \log(x^2 + 3x + 2) - \log((x + 1)^2) \]

Apply Logarithm Subtraction Rule

Use the subtraction rule of logarithms, which states that \(\log(A) - \log(B) = \log(\frac{A}{B})\). Applying this rule, the expression becomes: \[ \log\left(\frac{x^2 + 3x + 2}{(x + 1)^2}\right) \]

Simplify the Expression Inside the Logarithm

Factorize the numerator \(x^2 + 3x + 2\). It becomes \((x + 1)(x + 2)\). After substitution, the expression is: \[ \log\left(\frac{(x + 1)(x + 2)}{(x + 1)^2}\right) \]\Simplify the fraction by cancelling \(x + 1\) in the numerator and denominator: \[ \log\left(\frac{x + 2}{x + 1}\right) \]

Key concepts

logarithm power rule

The power rule of logarithms is a handy tool to simplify logarithmic expressions when you have an exponent inside the log function. The rule states that: \[\text{a} \log(b) = \log(b^{\text{a}}) \] This essentially means that when you have a coefficient multiplied with a logarithm, you can pull the coefficient up as the exponent of the argument inside the logarithm.
It's like taking the outside number and moving it as the power to the stuff inside the log.
In our exercise, we use this rule for the term \(-2 \log(x + 1)\). We turn it into \(\text{log}((x + 1)^2)\).
Remember, this helps in combining multiple logarithmic terms for further simplification.

logarithm subtraction rule

The subtraction rule or quotient rule of logarithms makes it easier to handle the difference of two logarithmic expressions. It states that: \[\text{log}(A) - \text{log}(B) = \text{log}\bigg(\frac{A}{B}\bigg)\] This means if you have the logarithm of a number minus the logarithm of another number, you can combine them into a single log by dividing the numbers inside the logs.
This rule is derived from the properties of exponents and allows us to simplify our expressions further by performing subtraction directly within the log function.
In our example, taking \(\text{log}(x^2 + 3x + 2) - \text{log}((x + 1)^2)\) and applying the subtraction rule, we convert it into one single logarithm: \(\text{log}\bigg(\frac{x^2 + 3x + 2}{(x + 1)^2}\bigg)\).

algebraic simplification

Algebraic simplification is all about breaking down and restructuring mathematical expressions to make them easier to understand and solve.
The goal is to reduce a complex expression into its simplest form by applying various algebraic techniques like factoring, cancelling common factors, and simplifying fractions.
In the given problem, after applying the logarithm rules, we are left with the term \(\text{log}\bigg(\frac{x^2 + 3x + 2}{(x + 1)^2}\bigg)\). The next step is to simplify the fraction inside the log.
The expression \((x^2 + 3x + 2)\) can be factored into \((x + 1)(x + 2)\).
Substituting this back, we obtain: \(\text{log}\bigg(\frac{(x + 1)(x + 2)}{(x + 1)^2}\bigg)\).
We can cancel out the common factor of \((x + 1)\), leaving us with \(\text{log}\bigg(\frac{x + 2}{x + 1}\bigg)\). Thus, we have successfully simplified the expression.