Exponential and Logarithm Function

An exponential function can be defined as following: For all real numbers x and for all positive real numbers b, b ¹ 1, the equation y = b^x, defines an exponential function with base b. There are two types of functions called increasing functions and decreasing functions. F is an increasing function if, for each a and b in the domain of f, whenever a > b, f(a) > f(b). F is a decreasing function if for each a and b in the domain of f , whenever a > b, f(a) < f(b). Two exponential functions are equal if and only if x = y all positive real numbers b, b ¹ 1.

There is another way to represent an exponential function. It is called the logarithm function. The logarithm function can be defined as following: For all positive real numbers x and b, b ¹ 1, there is a real number y such that y = log_bx if and only if x = b^y. To evaluate a logarithm function, you need to use the logarithm table or a calculator.

Here are some most common properties of logarithmic functions: For all positive real numbers b, b ¹ 1, and all admissible real values of x, y, m, and n,

        If log_bx = log_by, then x = y
        log_bb^x = x
        b^logbx= x, x> 0
        log_bb = 1
        log_b (mn) = log_bm + log_bn
        log_b (m/n) = log_bm – log_bn
        log _b (mⁿ) = n(log_bm)

For all positive real numbers a and b (except 1) and any positive real number x, log_ax = log_bx / log_ba.

There is a special type of logarithm called the natural logarithm. The natural logarithm is the logarithm with base-e. Natural logarithm is represented as In x.

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