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Inverse operations are fundamental in mathematics. They are operations that undo each other. For example, addition and subtraction are inverse operations. If you add a number and then subtract the same number, you get back to your starting value. Similarly, multiplication and division are inverses.
In the context of logarithms and exponents, these operations are also inverses of each other. If you start with an exponential function like \(10^2 = 100\), taking the logarithm base 10 of 100 would return you back to 2: \( \text{log}_{10}(100) = 2 \). This relationship helps in solving different types of equations where either logarithms or exponents appear.

A logarithm answers the question: To what exponent must we raise a given base to get a certain number? For instance, if you have \(10^2 = 100\), then \( \text{log}_{10}(100) = 2\). This tells you that 10 raised to the power of 2 equals 100.
Logarithmic functions are particularly useful for solving equations where the variable is an exponent. They appear in many scientific fields such as biology, astronomy, and economics.
Common properties of logarithms include:

• \( \text{log}_a(b \times c) = \text{log}_a(b) + \text{log}_a(c) \)
• \( \text{log}_a(\frac{b}{c}) = \text{log}_a(b) - \text{log}_a(c) \)
• \( \text{log}_a(b^c) = c \times \text{log}_a(b) \)
• \( \text{log}_a(a) = 1 \)
• \( \text{log}_a(1) = 0 \)

These properties can simplify complex calculations and are powerful tools in algebra and calculus.

Exponential functions involve constant rates of growth or decay. They have the form \( f(x) = a \times b^{x} \), where 'a' is a constant, 'b' is the base, and 'x' is the exponent. In such functions, the variable appears in the exponent.
One of the most common bases is 10, especially in scientific notation. For instance, \(10^3 = 1000\), which means 10 raised to the power of 3 is 1000.
Exponential functions are the opposite of logarithmic functions. Transforming a logarithm back into an exponential form can simplify many problems. If you have \( \text{log}_{10}(1000) = 3 \), you can rewrite it as \(10^3 = 1000\).
Exponential functions model many real-world phenomena:

• Population growth
• Radioactive decay
• Interest calculation in finance
• Growth of bacteria

Understanding exponential functions is crucial in various fields like physics, finance, and computer science.