In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(5x) + ln 1 = ln(5x)

Verified step by step guidance

1. Recall the logarithmic property that states: $\ln \left(a\right)+\ln \left(b\right)=\ln (a\cdot b)$. This means the sum of two natural logarithms is the natural logarithm of the product of their arguments.
2. Apply this property to the left side of the equation: $\ln \left(5x\right)+\ln \left(1\right)=\ln (5x\cdot 1)=\ln \left(5x\right)$.
3. Recognize that $\ln \left(1\right)=\; 0$ because the natural logarithm of 1 is always zero.
4. Since the left side simplifies to $\ln \left(5x\right)$, and the right side is also $\ln \left(5x\right)$, the equation is true as written.
5. Therefore, no changes are necessary to make the statement true.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the product rule: ln(a) + ln(b) = ln(ab). Understanding these rules helps in combining or breaking down logarithmic expressions correctly.

Logarithm of One

The natural logarithm of 1, ln(1), is always 0 because e^0 = 1. Recognizing this fact is essential when simplifying logarithmic expressions involving ln(1).

Equation Verification and Simplification

To determine if an equation involving logarithms is true, simplify both sides using logarithmic properties and evaluate constants. This process helps verify the equality or identify necessary corrections.