Logarithmic Functions Section 11.4 Logarithmic Functions

Objectives Define logarithm Write logarithmic equations as exponential equations Write exponential equations as logarithmic equations Evaluate logarithmic expressions Graph logarithmic functions Use logarithmic formulas and functions in applications

Objective 1: Define Logarithm Definition of Logarithm: For all positive numbers b, where b ≠ 1, and all positive numbers x, y = logb x is equivalent to x = by Because of this relationship, a statement written in logarithmic form can be written in an equivalent exponential form, and vice versa. The following diagram will help you remember the respective positions of the exponent and base in each form. we define the symbol log2 x to mean the power to which we raise 2 to get x.

Objective 2: Write Logarithmic Equations as Exponential Equations The following table shows the relationship between logarithmic and exponential notation. We need to be able to work in both directions.

EXAMPLE 1 Write each logarithmic equation as an exponential equation: Strategy To write an equivalent exponential equation, we will determine which number will serve as the base and which will serve as the exponent. Why We can then use the definition of logarithm to move from one form to the other:

EXAMPLE 1 Write each logarithmic equation as an exponential equation: Solution

EXAMPLE 1 Write each logarithmic equation as an exponential equation: Solution

Objective 3: Write Exponential Equations as Logarithmic Equations Write each exponential equation as a logarithmic equation:

EXAMPLE 2 Write each exponential equation as a logarithmic equation: Strategy To write an equivalent logarithmic equation, we will determine which number will serve as the base and where we will place the exponent. Why We can then use the definition of logarithm to move from one form to the other:

EXAMPLE 2 Write each exponential equation as a logarithmic equation: Solution

Objective 4: Evaluate Logarithmic Expressions We have seen that the logarithm of a number is an exponent. In fact, logb x is the exponent to which b is raised to get x. Translating this statement into symbols, we have

Objective 4: Evaluate Logarithmic Expressions For computational purposes and in many applications, we will use base-10 logarithms (also called common logarithms). When the base b is not indicated in the notation log x , we assume that b = 10: log x means log10 x The table below shows relation between base-10 logarithmic notation and exponential notation. In general, we have log1010x = x

EXAMPLE 4 Evaluate each logarithmic expression: Strategy After identifying the base, we will ask “To what power must the base be raised to get the other number?” Why That power is the value of the logarithmic expression.

EXAMPLE 4 Evaluate each logarithmic expression: Solution An alternate approach for this evaluation problem is to let log8 64 = x. When we write the equivalent exponential equation 8x = 64, it is easy to see that x = 2.

EXAMPLE 4 Evaluate each logarithmic expression: Solution We could also let The equivalent exponential equation is Thus, x must be –1.

EXAMPLE 4 Evaluate each logarithmic expression: Solution We could also let log4 2 = x. Then the equivalent exponential equation is 4x = 2. Thus, x must be

Objective 5: Graph Logarithmic Functions Logarithmic Functions: If b > 0 and b ≠ 1, the logarithmic function with base b is defined by the equations ƒ(x) = logb x or y = logb x The domain of ƒ(x) = logb x is the interval (0, ∞) and the range is the interval (–∞, ∞).

Objective 5: Graph Logarithmic Functions Properties of Logarithmic Functions The graph of ƒ(x) = logb x ( or y = logb x) has the following properties. It passes through the point (1, 0). It passes through the point (b, 1). The y-axis (the line x = 0) is an asymptote. The domain is the interval (0, ∞) and the range is the interval (–∞, ∞).

Objective 5: Graph Logarithmic Functions The exponential and logarithmic functions are inverses of each other, so their graphs have symmetry about the line y = x. The graphs of ƒ(x) = logb x and g(x) = bx are shown in figure (a) when b > 1 and in figure (b) when 0 < b < 1. The graphs of many functions involving logarithms are translations of the basic logarithmic graphs.

EXAMPLE 7 Graph each function by using a translation: Strategy We will graph g(x) = 3 + log2 x by translating the graph of ƒ(x) = log2 x upward 3 units. We will graph g(x) = log1/2 (x – 1) by translating the graph of ƒ(x) = log1/2 x to the right 1 unit. Why The addition of 3 in g(x) = 3 + log2x causes a vertical shift of the graph of the base-2 logarithmic function 3 units upward. The subtraction of 1 from x in g(x) = log1/2 (x – 1) causes a horizontal shift of the graph of the logarithmic function 1 unit to the right.

EXAMPLE 7 Graph each function by using a translation: Solution a. The graph of g(x) = 3 + log2 x will be the same shape as the graph of ƒ(x) = log2 x, except that it is shifted 3 units upward. See figure (a) below.

EXAMPLE 7 Graph each function by using a translation: Solution b. The graph of g(x) = log1/2 (x – 1) will be the same shape as the graph of ƒ(x) = log1/2 x, except that it is shifted 1 unit to the right. See figure (b) below.

Objective 6: Use Logarithmic Formulas and Functions in Applications Logarithmic functions, like exponential functions, can be used to model certain types of growth and decay. Common logarithm formula is used in electrical engineering to express the gain (or loss) of an electronic device such as an amplifier as it takes an input signal and produces an output signal. The unit of gain (or loss), called the decibel, which is abbreviated dB. Decibel Voltage Gain If Eo is the output voltage of a device and EI is the input voltage, the decibel voltage gain of the device (dB gain) is given by

EXAMPLE 9 Strategy We will find f(5). Stocking Lakes. To create the proper environmental balance, 250 hybrid bluegill were introduced into a lake by the local Fish and Game Department. Department biologists found that the number of bluegill in the lake could be approximated by the logarithmic function f(t) = 250 + 400 log(t + 1), where t is the number of years since the lake was stocked. Find the bluegill population in the lake after 5 years. Strategy We will find f(5). Why Since the variable t represents the time since the lake was stocked, t = 5.

EXAMPLE 9 Stocking Lakes. To create the proper environmental balance, 250 hybrid bluegill were introduced into a lake by the local Fish and Game Department. Department biologists found that the number of bluegill in the lake could be approximated by the logarithmic function f(t) = 250 + 400 log(t + 1), where t is the number of years since the lake was stocked. Find the bluegill population in the lake after 5 years. Solution

EXAMPLE 9 Stocking Lakes. To create the proper environmental balance, 250 hybrid bluegill were introduced into a lake by the local Fish and Game Department. Department biologists found that the number of bluegill in the lake could be approximated by the logarithmic function f(t) = 250 + 400 log(t + 1), where t is the number of years since the lake was stocked. Find the bluegill population in the lake after 5 years. Solution There were approximately 561 bluegill in the lake 5 years after it was stocked.