ex and In x

ex and In x

Basic Definitions of Exponentials and Logarithms

• The function e^x is called the exponential function. It’s so named because the variable, x, is the exponent.
• The function ln x is called the natural logarithm function. The term “natural” refers to the function’s base, which is the constant e.

Properties of Exponential Functions

• e^0 = 1: Any exponential function raised to the power 0 gives 1.
• e^1 = e: Any exponential function raised to the power 1 gives itself.
• e^(x + y) = e^x * e^y: This property is used when you need to simplify expressions with the same base and different powers.

Properties of Natural Logarithms

• ln 1 = 0: The logarithm of 1 to any base equals 0.
• ln e = 1: The natural log of e equals 1.
• ln(e^x) = x: This property is the fundamental property of logarithms and is crucial to solving exponential equations using natural logs.

Creating Equations and Solving Problems: Exponential Functions

• To formulate an exponential equation, remember that the basic form is y = ab^x where a is the initial amount, b is the growth factor and x is the exponent.
• Exponential functions can be used to model many real-world scenarios, including population growth, radioactive decay, and compound interest.

Creating Equations and Solving Problems: Logarithmic Functions

• The logarithmic form of an exponential equation is log_b(a) = x, where b is the base, a is the number you’re taking the log of, and x is the power the base must be raised to achieve a.
• Logarithmic functions can be applied to solve various real-world problems such as calculating pH levels, decibel levels, and Richter scale measurements.