Logarithms 

A logarithm is the inverse of an exponent. It tells us what power we must raise a base to, in order to get a given number.

Definition: If \( b^x = a \), then \( \log_b a = x \)

This reads as: “log base \( b \) of \( a \) is \( x \)”, meaning \( b \) raised to the power \( x \) gives \( a \).

 Vocabulary:

• Base: The number we raise to a power.
• Argument: The number we’re taking the log of.
• Logarithmic Form: \( \log_b a = x \)
• Exponential Form: \( b^x = a \)

Laws of Logarithms

Logarithmic expressions follow certain rules, much like exponent rules:

Important Properties

• \( \log_b 1 = 0 \) (because \( b^0 = 1 \))
• \( \log_b b = 1 \) (because \( b^1 = b \))
• \( \log_b b^x = x \)
• \( b^{\log_b x} = x \)

Graph of a Logarithmic Function

The function \( y = \log_b x \) is the inverse of \( y = b^x \)

• The graph passes through (1, 0)
• It has a vertical asymptote at \( x = 0 \)
• It increases slowly for larger \( x \)
• Domain: \( x > 0 \)
• Range: All real numbers (\( \mathbb{R} \))

Example base 10 graph: \( y = \log_{10} x \)

Example base e (natural log): \( y = \ln x \) – same rules but curved differently

Using Technology (Calculator)

Most calculators can compute:

• \( \log_{10} x \) (common log)
• \( \ln x = \log_e x \) (natural log)

To find \( \log_2 5 \), use:

\( \log_2 5 = \frac{\log 5}{\log 2} \approx \boxed{2.32} \)

Applications of Logarithms

• Used to solve exponential equations: \( 2^x = 17 \Rightarrow x = \log_2 17 \)
• Found in formulas involving growth/decay, earthquakes (Richter scale), sound (decibels), etc.