Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

Logarithmic Functions

For x > 0 and 0 < ≠,
y = log_ax if and only if x = ^y

The function given by f(x) = log_ax is called the logarithmic function with base .

Every logarithmic equation has an equivalent exponential form: y = log_ax is equivalent to x = ^y

A logarithmic function is the inverse function of an exponential function.

Exponential function: y=^x

Logarithmic function: y=log_ax is equivalent to x=^y

Examples: Write the equivalent exponential equation and solve for y.

Logarithmic Equation	Equivalent Exponential Equation	Solution
y = log₂16	16 = 2^y	16 = 2⁴ y = 4
y=log₃9	9=3^y	9 = 3² y = 2
y = log₄2	2 = 4^y	2 = 4^1/2 y = 1/2
y = log₄0	0 = 4^y	There is no power that 4 can be raised to obtain 0.
y = log₅1	1 = 5^y	1 = 5⁰ y = 0

The base 10 logarithm function f(x)=log₁₀x is called the common logarithm function.

Examples: 1) Calculate the values without using a calculator.

Function	Equivalent Exponential Equation	Solution
y = log₁₀100	100 = 10^y	100 = 10² y = 2
y = log₁₀0.01	0.01 = 10^y	1/100 = 10^-2 y = -2
y = log₁₀1/10	1/10 = 10^y	1/10 = 10^-1 y = -1
y = log₁₀10000	10000 = 10^y	10000 = 10⁴ y = 4

Properties of Logarithms

log_a1 = 0 since ⁰ = 1
log_a = 1 since ¹ =
log_a^x = x and ^logax = x inverse property
If log_ax = log_ay, then x = y one-to-one property

Examples:

2) Solve for x: log₆6 = x

log₆6 = 1 property 2 x = 1

3) Simplify: log₃3⁵

log₃3⁵ property 3

4) Simplify: 7^log₇9

7^log₇9 = 9 property 3

5) Graph f(x)=log₂x

Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function 2^x in the line y= x.

x	2^x
-3	1/8
-2	1/4
-1	1/2
0	1
1	2
2	4
3	8

2^x versus log2x

6) Graph the common logarithm function f(x)=log₁₀x.

x	1	2	4	10	100	1000
f(x)=log₁₀x	0	0.301	0.602	1	2	3

2^x versus log2x

The graphs of logarithmic functions are similar for different values of .

f(x) = log_ax ( > 1)

Domain: (0, infinite)
Range: (- infinite, infinite)
x-intercept: (1, 0)
vertical asymptote: x=0 or the y-axis as x-> 0⁺ and f(x) -> - Infinite
increasing (moves up to the right)
continuous
one-to-one
reflection of y = ^x in y = x

Graph of Log x

The function defined by f(x)=log_ex = ln x (x > 0, e 2.718281...) is called the natural logarithm function.

y = ln x is equivalent to e^y = x

Y=In x

Properties of Natural Logarithms

ln 1 = 0 since e⁰ = 1.
ln e = 1 since e¹ = e.
ln e^x = x and e^{ln x} = x inverse property
If ln x = ln y, then x = y. one-to-one property

Examples: Simplify each expression

7) ln 1/e² - ln e^-2 = -2 inverse property

8) e^ln20 = 20 inverse property

9) 3 ln e = 3(1) = 3 property 2

10) √ln 1 = √0 = 0 property 1

11) Carbon Dating: the formula (t in years) is used to estimate the age of organic material. The ratio of carbon 14 to carbon 12 in a piece of charcoal found at an archaeological dig is R = 1/10¹⁵. How old is it?

To the nearest thousand years the charcoal is 57,000 years old.

Change of Base Formula:

Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then, log_ax is given as follows:

log_ax = log_bx/log_b or log_ax = ln x/ln

Examples: 12)

Use common logarithms to evaluate log₃5.
log₃5 = ^log₁₀5⁄_log₁₀3

Use natural logarithms to evaluate log₆2.
log₆2 = ^{ln 2}⁄_{ln 6}

Properties of Logarithms:

Exponential property: ⁰ = 1

Corresponding log arithmic property: log_a1 = 0

Base a	Natural Base
a^maⁿ = a^m+n	e^meⁿ = e^m+n
a^m/aⁿ = a^{m - n}	e^m/eⁿ = e^{m - n}
(a^m)ⁿ = a^mn	(e^m)ⁿ = e^mn

Let a be a positive real number such that a ≠ 1, and let n be a real number. If u and v are real numbers, variables, or algebraic expressions such that u > 0 and v > 0, the following properties are true.

	Logarithm with Base a	Natural Logarithm
Product Property	log_a(uv) = log_au + log_av	ln (uv) = ln u + ln v
Quotient Property	log_au/v = log_au - log_av	ln u/v = ln u - ln v
Power Property	log_auⁿ = nlog_au	lnuⁿ = nlnu