2.6 Logarithmic Functions Inverse Functions, Summaries of Elementary Mathematics

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2.6 Logarithmic Functions

Inverse Functions

Question: What is the relationship between f (x) = x^3 and g(x) = √ (^3) x?

Question:√ What is the relationship between f (x) = x^2 and g(x) = x?

Definition (One-to-One Function)

A function f is said to be one-to-one if each range value corresponds to exactly one domain value.

Definition (Inverse of a Function)

If f is a one-to-one function, then the inverse of f is the function formed by interchanging the independent and dependent variables for f. Thus, if (a, b) is a point on the graph of f , then (b, a) is a point on the graph of the inverse of f.

Note: If f is not one-to-one, then f does not have an inverse.

So g(x) = 3

x is the inverse of f (x) = x^3 , and vice versa. However, h(x) = x^2 does not have an inverse, since it is not one-to one.

Another examples are

Definition (Logarithmic Function)

The inverse of an exponential function is called a logarithmic func- tion. For b > 0 and b 6 = 1,

y = logb x is equivalent to x = by

The log to the base b of x is the exponent to which b must be raised to obtain x.

Domain and Range of Logarithmic Function The domain of the logarithmic function is the set of all positive real numbers, which is also the range of the corresponding expo- nential function; and the range of the logarithmic function is the set of all real numbers, which is also the domain of the corresponding exponential function.

− 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6

− 5

− 3

− 1

1

3

5

7

9

x

y y = bx

x = by

y = x

− 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6

− 5

− 3

− 1

1

3

5

7

9

x

y y = bx

x = by

y = x

Example 1 Change each logarithmic form to an equivalent exponential form:

(a) log 3 9 = 2

(b) log 4 2 = (^12)

(c) log 3 19 = − 2

Example 2 Change each exponential form to an equivalent logarithmic form:

(a) 49 = 7^2

(b) 6 =

(c) 13 = 3−^1

Example 3 Find y, b, or x, as indicated

(a) Find y: y = log 9 27

(b) Find x: log 3 x = − 1

(c) Find b: logb 1000 = 3

Example 4

(a) logb (^) yzx =

(b) loga

(w v

(c) 2u^ log^2 b^ =

(d) log log^22 xb =

Example 5 Solve for x:

(a) 3 logb 2 + 12 logb 25 − logb 20 = logb x

(b) log 3 x + log 3 (x − 3) = log 3 10

Applications

Example 6 (Doubling Time for an Investment) How long will it take money to double if it is invested at 13% com- pounded annually?