Exponential & Logarithmic Functions

Learning Objectives Upon completing this module, you should be able to:  Define Logarithmic and Exponential functions . Understand the inverse relation between Logarithmic and Exponential functions. Know that Logarithmic and Exponential functions can be used to model a variety of real world situations. Use the properties of logarithms and exponents for rewriting expressions and solving equations, and graphing functions Find the derivatives of Logarithmic and Exponential functions .

… Population growth is a classic example Geometric growth 32 64 8 16 4 5 6

Exponentials f(x) = ax, a > 0 exponential increase 0 < a < 1 exponential decrease As x becomes very negative, f(x) gets close to zero As x becomes very positive, f(x) gets close to zero

f(x) = ax is one-to-one. For every x value there is a unique value of f(x). This implies that f(x) = ax has an inverse. f-1(x) = logax, logarithm base a of x.

f(x) = ax f(x) = logax

logax is the power to which a must be raised to get x. y = logax is equivalent to ay = x f(f-1(x)) = alogax = x, for x > 0 f-1(f(x)) = logaax = x, for all x. There are two common forms of the log fn. a = 10, log10x, commonly written a simply log x a = e = 2.71828…, logex = ln x, natural log. logax does not exist for x ≤ 0.

PROPERTIES OF EXPONENTIAL FUNCTIONS Let f (x) = ax, a > 0 , a ≠ 1. 1. The domain of f (x) = ax is (–∞, ∞). 2. The range of f (x) = ax is (0, ∞); the entire graph lies above the x-axis. 3. For a > 1, Exponential Growth (i) f is an increasing function, so the graph rises to the right. (ii) as x → ∞, y → ∞. (iii) as x → –∞, y → 0.

For 0 < a < 1, - Exponential Decay (i) f is a decreasing function, so the graph falls to the right. (ii) as x → – ∞, y → ∞. (iii) as x → ∞, y → 0. 5. The graph of f (x) = ax has no x-intercepts, so it never crosses the x-axis. No value of x will cause f (x) = ax to equal 0.

THE VALUE OF e As n gets larger and larger, gets closer and closer to a fixed number. This irrational number is denoted by e and is sometimes called the Euler number. The value of e to 15 places is e = 2.718281828459045. 11

Or

Properties of Exponents OR

x = loga b ax = b base

Definition:

Rules of Logarithms Rules of Logarithms with Base a If M, N, and a are positive real numbers with a ≠ 1, and x is any real number, then 1. loga(a) = 1 2. loga(1) = 0 3. loga(ax) = x 4. 5. loga(MN) = loga(M) + loga(N) 6. loga(M/N) = loga(M) – loga(N) 7. loga(Mx) = x · loga(M) 8. loga(1/N) = – loga(N) These relationships are used to solve exponential or logarithmic equations

Changing the base of a logarithm lo gac = x → c ≡ ax so logbc ≡ logbax ≡ x·logba

Therefore or

COMMON LOGARITHMS The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: log x = log10 x. Thus, y = log x if and only if x = 10 y. Applying the basic properties of logarithms log 10 = 1 log 1 = 0 log 10x = x 19

NATURAL LOGARITHMS The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = loge x. Thus, y = ln x if and only if x = e y. Applying the basic properties of logarithms ln e = 1 ln 1 = 0 log ex = x 20

MODEL FOR EXPONENTIAL GROWTH OR DECAY A(t) = amount at time t A0 = A(0), the initial amount k = relative rate of growth (k > 0) or decay (k < 0) t = time 21

Example: Radioactive decay A radioactive material decays according to the law N(t)=5e-0.4t When does N = 1? For what value of t does N = 1?

DOMAIN OF LOGARITHMIC FUNCTION Domain of y = loga x is (0, ∞) Range of y = loga x is (–∞, ∞) Logarithms of 0 and negative numbers are not defined. 23

GRAPHS OF LOGARITHMIC FUNCTIONS 24

Derivatives Memorize

Examples. f (x) = 5 ln x. f ‘ (x) = (5)(1/x) = 5/x f (x) = x5 ln x. Note: We need the product rule. f ‘ (x) = (x 5 )(1/x) + (ln x)(5x 4 ) = x 4 + (ln x)(5x 4) For you: Find dy/dx for y = x x

Examples. f (x) = ln (x 4 + 5) f ‘ (x) = f ‘ (x) = f (x) = 4 ln √x

Example.

Differentiating Logarithmic Expressions EXAMPLE Differentiate. SOLUTION This is the given expression. Differentiate.

Differentiating Logarithmic Expressions CONTINUED Distribute. Finish differentiating. Simplify.

General Derivative Rules Power Rule General Power Rule Exponential Rule General Exponential Derivative Rule Log Rule General Log Derivative Rule