Remark 3 Chebyshev polynomials are used extensively in approximating functions due to their properties that minimize errors. These polynomials are called equal ripple polynomials, since their maxima and minima alternate between 1 and −1.

1.4Logarithmic, Exponential and Hyperbolic Functions

Most logarithmic tables have tables for log10 x, loge x, ex and e−x because of their universal applications to scientific problems. The key relationship between logarithmic functions and exponential functions, using the same base, is that each one is an inverse of the other. For example, for base 10, we have

N = 10x if and only if x = log10 N.

We get two very interesting relations, namely

x = log10(10x) and N = 10(log10 N).

1.4. LOGARITHMIC, EXPONENTIAL AND HYPERBOLIC FUNCTIONS27

For base e, we get

x = loge(ex) and y = e(loge y).

If b > 0 and b 6= 1, then b is an admissible base for a logarithm. For such an admissible base b, we get

x = logb(bx) and y = b(logb y).

The Logarithmic function with base b, b > 0, b 6= 1, satisfies the following important properties:

1.logb(b) = 1, logb(1) = 0, and logb(bx) = x for all real x.

2.logb(xy) = logb x + logb y, x > 0, y > 0.

3.logb(x/y) = logb x − logb y, x > 0, y > 0.

4.logb(xy) = y logb x, x > 0, x 6= 1, for all real y.

5. (logb x)(loga b) = loga xb > 0, a > 0, b 6= 1, a 6= 1. Note that logb x = loga x.

loga b

This last equation (5) allows us to compute logarithms with respect to any base b in terms of logarithms in a given base a.

The corresponding laws of exponents with respect to an admissible base b, b > 0, b 6= 1 are as follows:

1. b0 = 1, b1 = b, and b(logb x) = x for x > 0.

2.bx × by = bx+y

3.bx = bx−y

by

4.(bx)y = b(xy)

Notation: If b = e, then we will express

logb(x) as ln(x) or log(x).

The notation exp(x) = ex can be used when confusion may arise.

The graph of y = log x and y = ex are reflections of each other through the line y = x.

1.4. LOGARITHMIC, EXPONENTIAL AND HYPERBOLIC FUNCTIONS29

=ln[(x + 1)3(2x − 3)3/4] − ln[(1 + 7x)1/3(2x + 3)3/2]

=ln(x + 1)3 + ln(2x − 3)3/4 − {ln(1 + 7x)1/3 + ln(2x + 3)3/2}

=3 ln(x + 1) + 34 ln(2x − 3) − 13 ln(1 + 7x) − 32 ln(2x + 3)

Example 1.4.2 Solve the following equation for x:

log3(x4) + log3 x3 − 2 log3 x1/2 = 5.

Using logarithm properties, we get

4 log3 x + 3 log3 x − log3 x = 5 6 log3 x = 5

x = ln(1/2) = − ln(2).

Example 1.4.4 Prove that for all real x, cosh2x − sinh2 x = 1.

=41[e2x + 2 + e−2x) − (e2x − 2 + e−2x)]

=41[4]

=1

Example 1.4.5 Prove that

(a)sinh(x + y) = sinh x cosh y + cosh x sinh y.

(b)sinh 2x = 2 sinh x cosh y.

Equation (b) follows from equation (a) by letting x = y. So, we work with equation (a).

(a)sinh x cosh y + cosh x sinh y = 12(ex − e−x) · 21(ey + e−y)

+21(ex + e−x) · 21(ey − e−y)

=41[(ex+y + ex−y − e−x+y − e−x−y)

+(ex+y − ex−y + e−x+y − e−x−y)]

=41[2(ex+y − e−(x+y)]

=21(e(x+y) − e−(x+y))

=sinh(x + y).

Example 1.4.6 Find the inverses of the following functions:

(a) sinh x (b) cosh x (c) tanh x

(a) Let y = sinh x = 21(ex − e−x). Then

p

Since ex > 0 for all x, ex = y + 1 + y2.

1.4. LOGARITHMIC, EXPONENTIAL AND HYPERBOLIC FUNCTIONS31

On taking natural logarithms of both sides, we get

p

x = ln(y + 1 + y2).

The inverse function of sinh x, denoted arcsinh x, is defined by

√

arcsinh x = ln(x + 1 + x2)

(b) As in part (a), we let y = cosh x and

2exy = 2ex · 12(ex + e−x) = e2x + 1 e2x − (2y)ex + 1 = 0

p

ex = 2y ± 4y2 − 4 2

p

ex = y ± y2 − 1.

We observe that cosh x is an even function and hence it is not one-to- one. Since cosh(−x) = cosh(x), we will solve for the larger x. On taking natural logarithms of both sides, we get

p p

x1 = ln(y + y2 − 1) or x2 = ln(y − y2 − 1).

We observe that

Thus, we can define, as the principal branch,

√

arccosh x = ln(x + x2 − 1), x ≥ 1

Therefore, the inverse of the function tanh x, denoted arctanhx, is defined by

Exercises 1.4

1. Evaluate each of the following

2.Prove each of the following identities

(a)sinh(x − y) = sinh x cosh y − cosh x sinh y

(b)cosh(x + y) = cosh x cosh y + sinh x sinh y

10. Solve each of the following equations for the independent variable.