Chapter 4.3 Logarithms

The previous section dealt with exponential function of the form y = a x for all positive values of a, where a ≠1.

Logarithms The horizontal line test shows that exponential functions are one-to-one, and thus have inverse functions.

Logarithms The equation defining the inverse function is found by interchanging x and y in the equation that defines the function.

Doing so with As the equation of the inverse function of the exponential function defined by y = a x. This equation can be solved for y by using the following definition.

The “log” in the definition above is an abbreviation for logarithm. Read log a x as “the logarithm to the base a of x.” By the definition of logarighm, if y = log a x, then the power to which a must be raised to obtain x is y, or x = a y.

Logarithmic Equations The definition of logarithm can be used to solve a logarithmic equation, and equation with a logarithm in at least one term, by first writing the equation in exponential form.

Solve the equation.

Exponential and logarithmic functions are inverses of each other. The graph of y = 2 x is shown in red in the figure.

The graph of the inverse is found by reflecting the graph of y = 2 x across the line y = x. The graph of the inverse function, defined by y = log 2 x, shown in blue, has the y-axis as a vertical asymptote.

Since the domain of an exponential function is th set of all real numbers, the range of a logarithmic function also will be the set of all real numbers.

In the same way, both the range of an exponential function and the domainof a logarithmic function are the set of all positive real numbers, so logarithms can be found for positive numbers only.

Graph the function

Since a logarithmic statement can be written as an exponential statement, it is not surprising that the properties of logarithms are based on the properties of exponents. The properties of exponents allow us to change the form of logarithmic statements so that products can be converted to sums, quotienst can be converted to differences, and powers can be converted to products.

Two additional properties of logarithms follow directly from the definition of log a 1=0 and log a a = 1

Proof To prove the product property let m = log b x and n = log b y log b x = m means b m = x log b y = n means b n = y

Now consider the product xy xy = b m b n xy = b m+n log b xy = m +n log b xy = log b x + log b y

Rewrite each expression. Assume all variables represent positive real numbers, with

Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠1 and b≠1.

There is no property of logarithms to rewrite a logarithm of a sum or difference. That is why, in Example 5 (a) was not written as log 3 x + log 3 2 The distributive property does not apply in a situation like this because log 3 (x+y) is one term; “log” is a function name, not a factor.

Assume that log 10 2 =.3010