Solve each equation. Give solutions in exact form. ln e^x - 2 ln e = ln e⁴

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Recall the properties of logarithms and exponents: \(\ln e^x = x\) and \(\ln e = 1\). Use these to simplify each term in the equation.

Rewrite the equation \(\ln e^x - 2 \ln e = \ln e^4\) as \(x - 2(1) = 4\) by applying the properties from step 1.

Simplify the left side to get \(x - 2 = 4\).

Solve the linear equation for \(x\) by adding 2 to both sides: \(x = 4 + 2\).

Express the solution in exact form as \(x = 6\).

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Properties of Logarithms

Logarithmic properties such as the product, quotient, and power rules allow simplification of expressions. For example, ln(a^b) = b ln(a), and ln(x) - ln(y) = ln(x/y). These rules help rewrite and combine logarithmic terms to solve equations efficiently.

Natural Logarithm and Exponential Functions

The natural logarithm (ln) is the inverse of the exponential function with base e. Understanding that ln(e^x) = x and e^(ln x) = x is crucial for solving equations involving ln and e, as it allows conversion between logarithmic and exponential forms.

Solving Logarithmic Equations

Solving logarithmic equations involves isolating the logarithm, applying logarithmic properties, and then exponentiating both sides to eliminate the logarithm. This process yields exact solutions, often expressed in terms of constants like e or integers.

Solve each equation. Give solutions in exact form. ln ex - 2 ln e = ln e4