Question

simplify each expression.

\\(\ln e^{3x} = \\)

\\(\ln e^{-2x} = \\)

\\(e^{\ln 4} = \\)

\\(e^{\ln 5x} = \\)

Explanation:

⚡ Using what you learned: Logarithms and Exponential Equations

Step 1: Simplify the first expression

The first expression is:

$$ \ln e^{7x} $$

Since the natural logarithm \(\ln(x)\) and the exponential function \(e^x\) are inverse operations, they undo each other:

$$ \ln e^{7x} = 7x $$

Step 2: Simplify the second expression

The second expression is:

$$ \ln e^{-2x} $$

Using the same inverse property:

$$ \ln e^{-2x} = -2x $$

Step 3: Simplify the third expression

The third expression is:

$$ e^{\ln 4} $$

Since \(e^x\) and \(\ln(x)\) are inverse functions:

$$ e^{\ln 4} = 4 $$

Step 4: Simplify the fourth expression

The fourth expression is:

$$ e^{\ln 3x} $$

Using the inverse property:

$$ e^{\ln 3x} = 3x $$

Answer:

• \(\ln e^{7x} = 7x\)
• \(\ln e^{-2x} = -2x\)
• \(e^{\ln 4} = 4\)
• \(e^{\ln 3x} = 3x\)