Section 7.1: Introduction to Logarithms

This section introduces logarithms as the inverse of exponential functions, defines common and natural logarithms, and explains their properties.

Example 1: Understanding Logarithms

Write \( \log_2 8 \) in exponential form and evaluate it.

Step 1: Convert to exponential: \( \log_2 8 = x \Rightarrow 2^x = 8 \)

Step 2: Solve for x: \( 2^3 = 8 \), so \( x = 3 \)

Evaluate \( \ln e^5 \).

Step 1: Recall that \( \ln a = \log_e a \)

Step 2: \( \ln e^5 = 5 \)

Evaluate \( \log_3 27 \)
Convert \( \log_5 125 = ? \) to exponential form and solve
Evaluate \( \ln e^2 \)
Write \( 10^4 = 10000 \) in logarithmic form
Find \( x \) if \( \log_2 x = 6 \)