Solve each equation. Give solutions in exact form. See Examples 5–9. ln 4x = 1.5

Key Concepts

Natural Logarithm

The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is used to solve equations involving exponential growth or decay. Understanding how to manipulate natural logarithms is essential for solving equations like 'ln(4x) = 1.5'.

Exponential Equations

Exponential equations are equations in which variables appear as exponents. To solve these equations, one often uses logarithms to isolate the variable. In the context of the given question, converting the logarithmic equation back to its exponential form is a key step in finding the solution.

Exact Solutions

Exact solutions refer to the precise values of variables without approximation. In the context of logarithmic and exponential equations, this means expressing the solution in terms of logarithms or fractions rather than decimal approximations. This is important for maintaining accuracy in mathematical solutions.