Problem 80 - Intermediate Algebra

Question

Write each logarithmic expression as one logarithm. See Example 7. $$ 3 \log _{b}(x+1)-2 \log _{b}(x+2)+\log _{b} x $$

Step-by-Step Solution

Answer

\(\log_b \left(\frac{x(x+1)^3}{(x+2)^2}\right)\)

Step 1: Apply Product Property

Start by using the product property of logarithms, which states that if you have \( ext{log}_b M + ext{log}_b N\), you can combine them into \( ext{log}_b(MN)\). Here, we will combine the terms, but notice only the last term \(\log_b x\) could be directly used in combination later. So, let’s use it in subsequent steps.

Step 2: Apply the Power Property

Use the power property of logarithms, which allows us to move coefficients to exponents: \(a \cdot \log_b M = \log_b (M^a)\). Thus, \(3 \cdot \log_b (x+1)\) becomes \(\log_b ((x+1)^3)\) and \(2 \cdot \log_b (x+2)\) becomes \(\log_b ((x+2)^2)\).

Step 3: Subtract Within the Logarithmic Expression

Use the quotient property of logarithms, which states \(\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)\). We apply this for the first two transformed expressions: \(\log_b((x+1)^3) - \log_b((x+2)^2) = \log_b \left(\frac{(x+1)^3}{(x+2)^2}\right)\).

Step 4: Combine All into One Logarithm

Now combine the result of Step 3 with \(\log_b x\) using the product property: \(\log_b \left(\frac{(x+1)^3}{(x+2)^2}\right) + \log_b x = \log_b \left(x \cdot \frac{(x+1)^3}{(x+2)^2}\right) = \log_b \left(\frac{x(x+1)^3}{(x+2)^2}\right)\).

Key Concepts

Product Property of Logarithms

When dealing with logarithmic expressions, the product property is a useful tool. This property tells us that adding two logarithms with the same base is equivalent to taking the logarithm of the product of their arguments. For example, when we have \(\log_b M + \log_b N\), it simplifies to \(\log_b(MN)\).
This property works often like multiplication does in arithmetic. It lets us merge separate logarithmic terms into a single expression.
In practice, this means if you have \(\log_b 2 + \log_b 3\), you're actually working with \(\log_b(2 \times 3) = \log_b 6\).
Utilizing this property simplifies complex logarithmic expressions by reducing the number of separate terms.

• Efficient for simplifying and solving equations.
• Helps condense tasks involving multiple logs into single expressions.

Remember, it's only applicable when the base of the logarithms is the same.

Power Property of Logarithms

The power property of logarithms is another powerful principle that helps simplify logarithmic expressions. This property allows us to simplify an expression by converting a multiplication factor into an exponent.
This transformation works as follows: If you have a coefficient in front of a logarithm, like \(a \cdot \log_b M\), you can express it as \(\log_b (M^a)\).

This property is particularly helpful when simplifying or solving equations that involve logarithms. Let's take the expression \(3 \cdot \log_b(x+1)\).
This can be rewritten as \(\log_b((x+1)^3)\), effectively moving the `3` as an exponent to the argument.

• Especially useful for simplifying expressions with coefficients.
• Transforms logarithmic equations into polynomial equations.

Remember, it can only be applied when the base across the expression remains constant.

Quotient Property of Logarithms

Lastly, the quotient property is another critical principle in the realm of logarithms. It provides a way to deal with the subtraction of logarithms.
The rule is: \(\log_b M - \log_b N\) becomes \(\log_b \left(\frac{M}{N}\right)\).
This allows us to take the difference of two logarithms and use them to form a single logarithmic expression by dividing their arguments.
Imagine having \(\log_b 8 - \log_b 2\). According to this property, it becomes \(\log_b \left(\frac{8}{2}\right) = \log_b 4\).
This is particularly helpful when you're faced with equations or expressions involving subtracted logs.

• Converts subtractions into simpler division within a single logarithm.
• Facilitates easier manipulation and simplification of complex expressions.

Like the product property, this applies only when both logarithms share the same base.