In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$lo{g}_b(x^3+y^3)=3lo{g}_{b}x+3lo{g}_{b}y$

Verified step by step guidance

1. Recall the logarithm property for products: $\log b_{}(x\backslash (\backslash cdot\backslash )y)=\log b_{}\left(x\right)+\log b_{}\left(y\right)$. This means logarithms turn multiplication inside the log into addition outside the log.
2. Notice that the expression inside the logarithm on the left side is $x^3+y^3$, which is a sum, not a product. The logarithm property for sums is not the same as for products.
3. The right side of the equation is $3\; \backslash (\backslash log\backslash )\_b\; x\; +\; 3\; \backslash (\backslash log\backslash )\_b\; y$, which can be rewritten using the power rule of logarithms as $\backslash (\backslash log\backslash )\_b\; (x^3)\; +\; \backslash (\backslash log\backslash )\_b\; (y^3)$. Using the product rule, this equals $\backslash (\backslash log\backslash )\_b\; (x^3\; y^3)$.
4. Since $\backslash (\backslash log\backslash )\_b\; (x^3\; +\; y^3)$ is not equal to $\backslash (\backslash log\backslash )\_b\; (x^3\; y^3)$, the original statement is false.
5. To make the statement true, replace the sum inside the logarithm with a product: $\backslash (\backslash log\backslash )\_b\; (x^3\; y^3)\; =\; 3\; \backslash (\backslash log\backslash )\_b\; x\; +\; 3\; \backslash (\backslash log\backslash )\_b\; y$.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the product rule: log_b(MN) = log_b M + log_b N, and the power rule: log_b(M^k) = k log_b M. Understanding these rules is essential to manipulate and evaluate logarithmic expressions correctly.

Sum of Cubes vs. Product

The expression x^3 + y^3 is a sum of cubes, which cannot be factored into a simple product of x and y terms. Since logarithm properties apply to products, not sums, recognizing the difference between sums and products is crucial to avoid incorrect application of log rules.

True or False Statements in Algebra

Determining the truth of algebraic statements involves verifying if the given equality holds for all valid values. If false, one must identify the error and correct it, often by applying the correct algebraic or logarithmic rules, ensuring the statement accurately reflects mathematical principles.