Question

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

Short answer

\frac{1}{2} )}

Step-by-step solution

- Apply the Power Rule

The first step is to apply the power rule of logarithms, which states that for any positive number \(a\) and any real number \(b\), \ \ \ \ \ \ \ \ will lead to the following expressions: \ \ \ Applying this to our equation: \ \

- Combine Using the Product Rule

Now that we have applied the power rule, combine the logarithms into a single expression using the product rule of logarithms, which states that for any positive numbers \(a\) and \(b\), \ \ \ \ \ meaning our expression becomes: \

Key concepts

Power rule for logarithms

The power rule for logarithms is a fundamental concept that helps simplify complex logarithmic expressions. It states that for any positive number base, \(a\), and any real number, \(b\), the expression \[ \log_b(a^c) = c \cdot \log_b(a) \] holds true. This allows us to move the exponent of the argument to the front as a multiplier.
For example, given \(\frac{1}{3} \log(x^3 + 1)\) and \(\frac{1}{2} \log(x^2 + 1)\), applying the power rule means rewriting them as \(\log((x^3 + 1)^{1/3})\) and \(\log((x^2 + 1)^{1/2})\).
Notice how \(1/3\) and \(1/2\) have become exponents of \((x^3 + 1)\) and \((x^2 + 1)\), making further simplification possible.

Product rule for logarithms

The product rule for logarithms is another important property used for combining different logarithmic expressions into a single logarithm. It states that for any positive numbers \(a\) and \(b\), the relationship \[ \log_b(a) + \log_b(b) = \log_b(ab) \] is valid.
This rule tells us that the sum of the logarithms of two numbers is equal to the logarithm of their product. Applying this rule to our simplified expressions from the power rule, we get:
\(\log((x^3+1)^{1/3}) + \log((x^2+1)^{1/2}) \) becomes \[ \log( ((x^3+1)^{1/3}) \cdot ((x^2+1)^{1/2})) \]
This allows the two logarithms to merge into a single logarithmic expression.

Combining logarithms

Combining logarithms involves using the product rule to merge multiple logarithmic expressions. After applying the power rule and rewriting the original terms, we use the product rule to combine them into a single logarithmic statement.
Given \( \log( ((x^3+1)^{1/3}) \cdot ((x^2+1)^{1/2}) )\), this simplifies the original expression \( \frac{1}{3} \log(x^3 + 1) + \frac{1}{2} \log(x^2 + 1)\) into a single, more streamlined logarithmic term. This single logarithm provides a simpler and more manageable form for calculations and further analysis.
Understanding how to use these rules effectively is crucial for solving logarithmic expressions efficiently. By mastering the power rule and the product rule, students can tackle a wide variety of logarithmic problems with greater ease.