What Are Logarithms

A logarithm is a way of asking the question: “What power should I raise the base to, in order to get this number?”

For example, if you know that \( 2^3 = 8 \), then the logarithm tells you: \( \log_2 8 = 3 \). Logs help us reverse exponential calculations.

The formal definition of a logarithm is:

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

This means:

• \(b\) is the base
• \(a\) is the number
• \(x\) is the exponent

The logarithm simply finds the exponent.

Logarithms make it easier to work with very large numbers, very small numbers, and repeated multiplication. They are used when:

• Numbers grow rapidly (exponential growth)
• We deal with scientific data (earthquakes, sound, pH value)
• We solve exponential equations
• We simplify multiplication into addition

Logs reduce big calculations into smaller, manageable steps.

Exponents and logarithms are inverse operations. Just like addition and subtraction undo each other, logs and powers undo each other.

Logarithms are written with different bases depending on use:

• Base 10 (common log): used in scientific calculations → written as log x
• Base e (natural log): used in calculus → written as ln x
• Base 2: used in computers and binary systems → written as log₂ x

• Earthquake magnitude uses logarithmic scale (Richter scale).
• Sound intensity uses decibels (dB), a log scale.
• pH value in chemistry is logarithmic.
• Computer science uses log base 2 frequently.
• Finance: compound interest grows exponentially, solved using logs.

• \( \log_2 8 = 3 \) because \( 2^3 = 8 \)
• \( \log_{10} 1000 = 3 \) because \( 10^3 = 1000 \)
• \( \log_5 25 = 2 \) because \( 5^2 = 25 \)
• \( \log_3 81 = 4 \) because \( 3^4 = 81 \)

• Thinking \( \log_b (x+y) = \log_b x + \log_b y \) (this is false).
• Using a negative or zero number inside the log (not allowed).
• Confusing base with argument.
• Assuming all logs mean base 10 (only 'log' without base implies 10).

Rewrite in exponential form:

1. \( \log_4 64 = 3 \)
2. \( \log_7 49 = 2 \)
3. \( \log_3 1 = 0 \)

Now convert these to log form:

1. \( 2^5 = 32 \)
2. \( 10^2 = 100 \)
3. \( 3^4 = 81 \)

• A logarithm finds the exponent for a given base.
• Definition: \( \log_b a = x \iff b^x = a \).
• Logs undo exponential calculations.
• They help simplify large numbers and solve exponential problems.
• Used widely in science, technology, and mathematics.