In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.ln x + ln 7

The properties of logarithms are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule, which states that the logarithm of a product is the sum of the logarithms (ln a + ln b = ln(ab)), and the power rule, which allows you to bring exponents in front of the logarithm (k * ln a = ln(a^k)). Understanding these properties is essential for condensing logarithmic expressions.

The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is approximately 2.71828. It is commonly used in mathematics, particularly in calculus and exponential growth models. Recognizing that ln x represents the power to which 'e' must be raised to obtain 'x' is crucial for evaluating and manipulating expressions involving natural logarithms.

Condensing logarithmic expressions involves combining multiple logarithms into a single logarithm. This process utilizes the properties of logarithms to simplify the expression, often resulting in a more manageable form. For example, the expression ln x + ln 7 can be condensed to ln(7x) by applying the product rule, which is a fundamental skill in algebra.