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Daniel Brown Final Project INTD 301
Understand the properties of exponential and logarithmic functions (0009)
I. Review exponents Let us review properties of exponents from algebra class. Any number b , where n is a natural number, are defined by:
bn^ = b · b · … · b (n-factors) Ex: 2^4 = 2 · 2 · 2 · 2 = 16 (Note: b^0 = 1. Any number with a power of zero is equal to one)
-Exponents involving negative powers:
b-n^ = 1/(bn) Ex: 2-6^ = 1/2^6 = 1/
-Exponents involving fractional powers: (where p and q are both natural numbers)
bp/q^ = qvbp^ = (qvb)p^ Ex: 84/3^ = 3 v8^4 = (^3 v8)^4 = 2^4 = 16
Let us also review our laws of exponents:
i) Multiplying terms with the same base, add their exponents bpbq^ = bp+q^ Ex: 5^2 · 5^8 = (5·5)(5·5·5·5·5·5·5·5) = 52+8^ = 5^10
ii) Dividing terms with the same base, subtract their exponents bp^ /bq^ = bp-q^ Ex: 5^2 /5^8 = (5·5) / (5·5·5·5·5·5·5·5) = 1/(5·5·5·5·5·5) = 52-8^ = 5-
iii) Powers raised to a power, multiply their exponents (bp)q^ = bp·q^ Ex: (5^2 )^8 = (5·5)(5·5)(5·5)(5·5)(5·5)(5·5)(5·5)(5·5) = 52·8^ = 5^16
II. Review Logarithms Let us also recall logarithms from algebra class, such that for any number b , where x >0: logbx = n
Furthermore, the above equation is defined to be that exponent ( n ) to which base ( b ) must be raised to produce x. Such that, bn^ = x
Ex: log 10 100 = 2 log 2 16 = 4 102 = 100 24 = 16
Common Logarithm: logarithms with base 10. It is standard to denote common logs by simply writing ‘log x,’ it is then implied it will be base 10.
Natural Logarithm: logarithms with base e. It is standard to denote natural logs by ‘ln x,’ instead of loge x. Ex: y = ln x x = e y (Note: ln e = 1, the logarithm with base of itself is always 1)
Similarly to the laws of exponents are the laws of logarithms:
i) The logarithm of a product is equal to the sum of the logarithms of each factor. logbxy = logbx + logby
ii) The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. logb(x/y) = logbx - logby
iii) The logarithm of a power of x is equal to the exponent of that power times the logarithm of x. logbxn^ = n·logbx
iv) The logarithm of a reciprocal of some number is equal to the negation of the logarithm of that number. logb(1/c) = - logbc
III. Analyze the relationship between logarithmic and exponential functions (Definitions found Calculus , Anton) -Lets first look at each function separately
A.) Exponential Functions : An exponential function is a function of the form f(x) = bx, where b is a positive constant and x is a variable exponent. Ex: f(x) = 2x, f(x) = (1/2)x, f(x) = px Not Ex: f(x) = x^2 , f(x) = xp
B.) Logarithmic Functions: A logarithmic function is a function of the form f(x) = logbx, where b is a positive constant not equal to one, and x>0.
We can obtain the formula for a logarithmic function from the formula of an exponential function by solving algebraically for an inverse. Ex: f(x) = bx y = bx. To solve for an inverse algebraically, we switch the x and y variables such that,
3.) (1/3)·ln x – ln(x^2 -1) + 2·ln(x+3) Again we are looking to condese the equation and combine all the terms. Thus we will first use the power property. ln x(1/3)^ – ln(x^2 -1) + ln(x+3)^2. Next, we will use the product property, ln (^3 vx)·(x+3)^2 – ln(x^2 -1). And lastly, the quotient rule, ln [(^3 vx)·(x+3)^2 /(x^2 -1)].
V. Analyzing the graphs of logarithmic or exponential functions or re lation (Graphs obtained from The Math Page: Topics in Precalculus , <http://www. themathpage.com/aPreCalc/logarithmic-exponential- functions.htm#exp>) Let us look at the graphs of each function separately.
A.) Exponential Graph : Where b >0, b? 1 and x? R
f(x) = bx^ is defined for all real values of x, so its domain is (-8, +8). f(x) = bx^ is continuous on the interval (-8, +8), so its range is (0, +8).
For the function f(x) = bx, when 0<b<1, we observe a reflection of the graph over the y- axis. As b gets closer to 0 it is decreasing at a faster then when b is approaching 1. When b>1, we observe the graph increasing from -8 towards +8. As b approaches +8, the function is increasing at a faster rate then when b is near 1. Visually, the graph appears to be steeper as b gets larger.
If b = 1 the function is constant. Since bx^ = 1x^ = 1 for every value of x.
B.) Logarithmic Graph :
Since logarithmic functions are the inverse of exponential functions, the graphs are a reflection over the line y = x. Since the graph of exponential function shows a rapidly increasing function (where b>1), its reflection (inverse) shows a slowly increasing function.
VI. Solving problems involving exponential growth and decay
Exponential Growth : Exponential growth refers to the rate in which something increases. If something is said to grow exponentially, then it is growing proportional to its size. Hence, the larger the quantity gets, the faster it will grow. See below, in the table, the larger the value of b, the faster it will reach +8.
Ex: x 2 x^10 x 1 2 10 2 4 100 3 8 1, 4 16 10, 5 32 100, 10 1,024 1,000,000,000, 100 ‡+8 ‡+
10 x^ has a larger quantity then 2x, therefore increases at a faster rate. When 2x^ reaches a quantity of 1,024, 10x^ has already reached a value of 10,000,000,000.
Exponential Decay : Exponential decay has the same concept as exponential growth. However, instead of an increasing function, it is decreasing. Exponential decay occurs when 0 < b < 1. The closer the value of b is to 0, the faster it will converge to 0
