Problem 48 - Precalculus

Question

Write each expression as a single logarithm. $$\frac{1}{3} \ln \left(x^{2}+4\right)-\frac{1}{2} \ln \left(x^{2}-3\right)-\ln (x-1)$$

Step-by-Step Solution

Answer

\( \ln\left(\frac{(x^2 + 4)^{1/3}}{(x^2 - 3)^{1/2}(x - 1)}\right) \) is the single logarithm.

Step 1: Apply the Power Rule

Use the power rule of logarithms, which states that \( a \log(b) = \log(b^a) \), to rewrite each logarithm. The expression becomes: \[ \ln((x^2 + 4)^{\frac{1}{3}}) - \ln((x^2 - 3)^{\frac{1}{2}}) - \ln(x - 1) \]

Step 2: Combine Using the Quotient Rule

Use the quotient rule of logarithms, \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \), to combine the terms. Start by combining the first two terms: \[ \ln\left(\frac{(x^2 + 4)^{\frac{1}{3}}}{(x^2 - 3)^{\frac{1}{2}}}\right) \] Then, include the third term using the quotient rule again: \[ \ln\left(\frac{(x^2 + 4)^{\frac{1}{3}}}{(x^2 - 3)^{\frac{1}{2}}} \cdot \frac{1}{x - 1}\right) = \ln\left(\frac{(x^2 + 4)^{\frac{1}{3}}}{(x^2 - 3)^{\frac{1}{2}}(x - 1)}\right) \]

Step 3: Write as a Single Logarithm

The expression is now written as a single logarithm. Final expression: \[ \ln\left(\frac{(x^2 + 4)^{\frac{1}{3}}}{(x^2 - 3)^{\frac{1}{2}}(x - 1)}\right) \]

Key Concepts

Power Rule of Logarithms

The power rule is an essential concept when dealing with logarithms, especially in simplifying expressions with fractional coefficients. It states that if you have a logarithm with a coefficient, you can rewrite it by using the base raised to the power of that coefficient. This means \( a \log(b) = \log(b^a) \). Make sure to apply this correctly to each logarithmic term in the expression.

For instance, in our exercise:

• We started with \( \frac{1}{3} \ln(x^2 + 4) \), which became \( \ln((x^2 + 4)^{\frac{1}{3}}) \).
• The term \( \frac{1}{2} \ln(x^2 - 3) \) was rewritten as \( \ln((x^2 - 3)^{\frac{1}{2}}) \).

By applying the power rule, each logarithm becomes an expression involving a power, which lays the groundwork for further simplification.

Quotient Rule of Logarithms

The quotient rule of logarithms helps us combine multiple logarithmic expressions into one. It states that the difference of two logarithms can be rewritten as the logarithm of the quotient of their arguments: \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \).

In our example, after applying the power rule, we used the quotient rule to combine the first two logarithms:

• Started with \( \ln((x^2 + 4)^{\frac{1}{3}}) - \ln((x^2 - 3)^{\frac{1}{2}}) \) and wrote it as \( \ln\left(\frac{(x^2 + 4)^{\frac{1}{3}}}{(x^2 - 3)^{\frac{1}{2}}}\right) \).

This was then combined with the final term:

• We added \( -\ln(x - 1) \) to get \( \ln\left(\frac{(x^2 + 4)^{\frac{1}{3}}}{(x^2 - 3)^{\frac{1}{2}}(x - 1)}\right) \).

By using the quotient rule, we efficiently reduced a complex expression into a single, elegant logarithm.

Combining Logarithms

Combining logarithms effectively is vital for simplification and solving equations. By transforming multiple logarithmic expressions into one single logarithm, you preserve the meaning of the expression while eliminating unnecessary complexity.

In this exercise, after applying both the power and quotient rules, we consolidated everything into one clean expression:

• We successfully expressed \( \ln\left(\frac{(x^2 + 4)^{\frac{1}{3}}}{(x^2 - 3)^{\frac{1}{2}}(x - 1)}\right) \).

Always ensure that each step logically follows from the previous one and establishes the rules of logarithms. This method not only simplifies the given expressions but also strengthens understanding and mastery over logarithmic operations.