Logarithmic functions are mathematical expressions involving logarithms, typically written in the form:

$y = \log_b(x)$

where $b$ is the base, $x$ is the argument, and $y$ is the result. Logarithmic functions are the inverses of exponential functions, which makes them particularly useful for solving equations where the variable is in the exponent.

Key Properties of Logarithmic Functions:

1. Inverse Property: $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$
2. Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
3. Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$
4. Power Rule: $\log_b(x^y) = y\log_b(x)$
5. Change of Base Formula: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$, where $k$ is any positive number.

Applications:

• Solving Exponential Equations: Logarithms allow us to bring down exponents and solve for variables in equations like $2^x = 16$.
• Modeling Growth and Decay: Used in natural phenomena such as population growth, radioactive decay, and sound intensity.
• Data Compression: Logarithms are used in algorithms like the JPEG format and in computer science.

Would you like me to demonstrate the use of logarithmic functions to solve an equation, graph them, or show their application in a specific context?

Questions to Expand Your Understanding:

1. How are logarithmic functions used to solve exponential equations in practice?
2. What is the difference between natural logarithms ($\ln$) and logarithms with other bases?
3. Can you derive the change of base formula for logarithms?
4. How do logarithmic functions behave graphically, and what are their asymptotes?
5. What are some real-life applications of logarithmic functions in science and engineering?

Tip:

When working with logarithmic functions, always remember that the argument ($x$) must be positive, as logarithms are undefined for non-positive values.