Solve each equation. Give solutions in exact form. See Examples 5–9. ln e^x - 2 ln e = ln e^4

Verified step by step guidance

1. Start by using the properties of logarithms to simplify the equation. Recall that \( \ln(a^b) = b \ln(a) \). Apply this to \( \ln(e^x) \) and \( \ln(e^4) \).
2. Simplify \( \ln(e^x) \) to \( x \ln(e) \) and \( \ln(e^4) \) to \( 4 \ln(e) \). Since \( \ln(e) = 1 \), this further simplifies to \( x \) and \( 4 \) respectively.
3. Substitute these simplifications back into the equation: \( x - 2 \ln(e) = 4 \).
4. Simplify \( 2 \ln(e) \) to \( 2 \times 1 = 2 \). The equation now becomes \( x - 2 = 4 \).
5. Solve for \( x \) by adding 2 to both sides of the equation, resulting in \( x = 6 \).

Key Concepts

Natural Logarithm (ln)

The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is a fundamental concept in algebra and calculus, particularly in solving equations involving exponential functions. Understanding properties of logarithms, such as ln(a*b) = ln(a) + ln(b) and ln(a^b) = b*ln(a), is essential for manipulating and solving logarithmic equations.

Properties of Exponents

Exponents represent repeated multiplication of a base number. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the power of a product ( (ab)^n = a^n * b^n). These properties are crucial for simplifying expressions and solving equations that involve exponential terms, such as e^x.

Solving Exponential Equations

Solving exponential equations involves isolating the variable in the exponent. This often requires applying logarithmic functions to both sides of the equation to bring the exponent down. For example, if you have e^x = a, taking the natural logarithm of both sides gives x = ln(a). Mastery of this technique is vital for finding exact solutions in problems involving exponential growth or decay.