Relation Between Exponential and Logarithmic Forms

Exponential and logarithmic expressions are closely related because logarithms are the inverse of exponential functions.

This means logs undo exponentiation, and exponentiation undoes logarithms. Understanding this relationship is the key to mastering logarithms.

The most important relationship is:

\( b^x = a \iff \log_b a = x \)

This allows us to switch between exponential and logarithmic forms easily.

Think of exponential form as building a number:

Example: \( 2^5 = 32 \)

Logarithmic form is about finding the exponent behind the number:

Example: \( \log_2 32 = 5 \)

They represent the same fact in two different ways.

Because they are inverse functions:

• \( b^{\log_b x} = x \)
• \( \log_b (b^x) = x \)

This is similar to how square roots and squaring undo each other.

• Exponential to Logarithmic:

\( 3^4 = 81 \iff \log_3 81 = 4 \)

• Logarithmic to Exponential:

\( \log_5 25 = 2 \iff 5^2 = 25 \)

The graphs of \( y = b^x \) and \( y = \log_b x \) are mirror images of each other across the line \( y = x \).

• If you know exponential properties, you can derive logarithmic laws.
• \( b^{x+y} = b^x b^y \iff \log_b (xy) = \log_b x + \log_b y \)
• \( b^{x-y} = \dfrac{b^x}{b^y} \iff \log_b \left( \dfrac{x}{y} \right) = \log_b x - \log_b y \)

• Writing \( \log_b a = a^b \) (incorrect).
• Mixing up base and power in conversion.
• Trying to convert logs with negative arguments (undefined).
• Assuming \( \log_b (x+y) \) follows log laws (it does not).

Convert each pair:

1. \( 7^3 = 343 \)
2. \( \log_4 64 = 3 \)
3. \( 2^0 = 1 \)
4. \( \log_9 81 = 2 \)

• Exponential and logarithmic forms describe the same relationship.
• Conversion rule: \( b^x = a \iff \log_b a = x \).
• They are inverse functions.
• Their graphs are mirror images across \( y = x \).