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.
log_b (xy)⁵ = (log_b x + log_b y)⁵

Verified step by step guidance

1. Recall the logarithm property for the logarithm of a product: $\log b_{}\left(xy\right)=\log b_{}x+\log b_{}y$. This means the log of a product is the sum of the logs.
2. Apply the power rule of logarithms: $\log b_{}(a{)}^{\mathrm{n}}=\; n\log b_{}a$. This means the exponent can be brought in front as a multiplier.
3. Rewrite the left side of the equation using these properties: $\log b_{}\left(xy{)}^5=\; 5\log b_{}\right(xy)$.
4. Next, use the product rule inside the logarithm: $5\log b_{}\left(xy\right)=\; 5\; (\log b_{}x+\log b_{}y)$.
5. Compare this with the right side of the original statement: $(\log b_{}x+\log b_{}y)^5$. Notice that the original statement raises the sum to the fifth power, which is different from multiplying the sum by 5. Therefore, the original statement is false, and the correct statement is $\log b_{}(xy{)}^5=\; 5\; (\log b_{}x+\log b_{}y)$.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the product rule: log_b(xy) = log_b(x) + log_b(y), and the power rule: log_b(x^n) = n * log_b(x). Understanding these rules is essential to manipulate and evaluate logarithmic expressions correctly.

Exponentiation and Logarithms

When an exponent is applied to a product inside a logarithm, the power rule allows the exponent to be factored out: log_b((xy)^5) = 5 * log_b(xy). This differs from raising the sum of logarithms to a power, which is not equivalent, highlighting the importance of correctly applying exponent rules in logarithmic contexts.

Distinguishing Between Expressions and Their Equivalents

It is crucial to recognize that (log_b x + log_b y)^5 is not the same as 5 * (log_b x + log_b y). The former raises the entire sum to the fifth power, while the latter multiplies the sum by 5. This distinction helps avoid common mistakes when simplifying or verifying logarithmic statements.