Solve each equation. Give solutions in exact form. ln 4x = 1.5

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Identify the given equation: $\ln 4x = 1.5$. This means the natural logarithm of \$4x$ equals $1.5$.

Recall that the natural logarithm function $\ln y$ is the inverse of the exponential function $e^y$. To solve for $x$, rewrite the equation in exponential form: $4x = e^{1.5}$.

Isolate $x$ by dividing both sides of the equation by 4: $x = \frac{e^{1.5}}{4}$.

Express the solution in exact form, which involves leaving the answer in terms of $e$ without approximating the value of $e^{1.5}$.

Check the solution by substituting $x$ back into the original equation to ensure the equality holds true.

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주요 개념

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Natural Logarithm (ln) Function

The natural logarithm, denoted as ln, is the inverse of the exponential function with base e. It answers the question: to what power must e be raised to get a given number? For example, ln(4x) = 1.5 means e raised to 1.5 equals 4x.

Solving Exponential and Logarithmic Equations

To solve equations involving logarithms, rewrite the equation in exponential form to isolate the variable. For ln(4x) = 1.5, rewrite as 4x = e^{1.5}, then solve for x by dividing both sides by 4.

Exact Form Solutions

An exact form solution expresses the answer without decimal approximations, often using constants like e or π. Here, the solution should be left as x = e^{1.5} / 4 rather than a decimal, preserving mathematical precision.