Solving Exponential and Logarithmic Equations
Exponential Equations (variable in exponent position)
1. Isolate the exponential portion ( base exp onent ): Move all nonexponential factors or terms to the other side of the equation. 2. Take ln or log of each side of the equation. • Make sure to use ln if the base is “e”. Then remember that ln e = 1 . • Make sure to use log if the base is 10. • If the base is neither “e” nor “10”, use either ln or log, your choice.. 3. Bring the power (exponent) down into coefficient position. 4. Use various algebra techniques to solve for the variable. 5. Check your answer by evaluating the original equation with your calculator.
Example: 4e 2 x −3 = 40
Answer is on next page
Logarithmic Equations
1. 2. 3. 4. 5. Move all log terms to one side of the equation, all nonlog terms to the other side. Combine log terms into a single log term using the laws of logarithms. Write the log equation in its exponential form. (remember: 2 3 = 8 ↔ log 2 8 = 3 ) Use various algebra techniques to solve for the variable. Check your answer using your calculator. Remember that domain problems occur in log functions. • If the base of the log is “10” or “e”, you can use the appropriate calculator keys. • If the log is not “10” or “e”, you may need to use the change of base formula before using your calculator.
Example:
log 5 ( x + 1) − 2 = log 5 ( x − 1)
Answer is on next page
Solving Exponential and Logarithmic Equations: Answers
Exponential Equation: 4e 2 x −3 = 40 Divide both sides by 4 to isolate the exponential portion
e 2 x −3 = 10
Take ln of both sides (use ln because the base is “e”) Use the law of logs that allows you to bring the power into coefficient position Recall that ln e is equivalent to 1, so the equation is actually → Now just solve for x using algebra (add 3 to both sides, then divide both sides by 2)
ln e 2 x −3 = ln 10 (2 x − 3) ln e = ln 10
2 x − 3 = ln 10
...EXPONENTIAL AND LOGARITHMIC FUNCTIONS
I.EXPONENTIAL FUNCTION
A. Definition
An exponential function is a function defined by f(x) = ax , where a > 0 and a ≠ 1. The domain of the function is the set of real numbers and the range is the set of positive numbers.
B. Evaluating Exponential Functions
1. Given: f(x) = 2x, find
a. f(3) = ____ b. f(5) = _____ c. f(2) = ______ d. f(4) = ______
2. Evaluate f(x) = ( 1)x if
2
a. x = 2 ____ b. x = 4 _____ c. x = 3 ______ d. x = 4 _______
C. Graphing Exponential Functions
On the same Cartesian Coordinate plane, sketch the graphs of each set of exponential function
1. a) f(x) = 2x b. f(x) = 3x c. f(x) = 4x
2. a) f(x) = 2x b. f(x) = 2x + 1 c. f(x) = 2x – 1
3. a) f(x) = 2x b. f(x) = 2x + 2 c. f(x) = 2x – 3
4. a) g(x) = 2 x b. g(x) = 3 x c. g(x) = 4 – x
D. The Property of Equality for ExponentialEquations
Let a, b and c be real numbers and a≠ 0, then ab = ac if and only if b = c
Examples:
1. 32x = 36 2. 23x = 8 3. 643x = 8 4. 10 x = 1/10000
5. 43x = 16x + 2 6. 16 x = 1/64 7. 93x = [ 1/3]5 8.5x+2 – 5x + 1 + 5x =
2625
Do as directed:
A. Evaluate:
1. If f(x) = 3x, what is
a. f( 3)?= ____ b. f( 4) = _____ c. f( 2) = _____ d. f( 4) = _____
2. What is g(x) = [ 1/3 ]x if
a.x = 2 ____ b....
...of functions are the exponential functions and the logarithmic functions. Exponential functions are the functions in the form of y = ax, where ''a'' is a positive real number, greater than zero and not equal to one. Logarithmic functions are the inverse of exponential functions, y = loga x, where ''a'' is greater to zero and not equal to one. These functions have certain differences as well as similarities between them. Also they are very useful for various situations in life.
Logarithmic functions are fairly different from the exponential functions. The first difference that we can find between them is in the equations, they are inverse to each other. The logarithmicequation is y = loga x and the exponentialequation is y = ax. We can also see that the natural exponential function is different form the natural logarithmic function. The natural exponential function is y = f(x) = ex and the natural logarithmic function is f(x) = loge x = lnx , where x > 0. Also we can see that to graph and exponential function it always has to pass through the point (0,1).
However, both of these functions also have similarities. Both of the functions contain an ''a'' which has to be greater than zero and less than one. Also when we graph both of the...
...MATH133 Unit 5: Exponential and Logarithmic Functions
Individual Project Assignment: Version 2A
Show all of your work details for these calculations. Please review this Web site to see how to
type mathematics using the keyboard symbols.
IMPORTANT: See Question 1 in Problem 2 below for special IP instructions. This is
mandatory.
Problem 1: Photic Zone
Light entering water in a pond, lake, sea, or ocean will be absorbed or scattered by the particles
in the water and its intensity, I, will be attenuated by the depth of the water, x, in feet. Marine
life in these ponds, lakes, seas, and oceans depend on microscopic plant life that exists in the
photic zone. The photic zone is from the surface of the water down to a depth in that particular
body of water where only 1% of the surface light remains unabsorbed or not scattered. The
equation that models this light intensity is the following:
𝐼 = 𝐼0 𝑒 −𝑘𝑥
In this exponential function, I0 is the intensity of the light at the surface of the water, k is a
constant based on the absorbing or scattering materials in that body of water and is usually called
the coefficient of extinction, e is the natural number 𝑒 ≅ 2.718282, and I is the light intensity at
x feet below the surface of the water.
1. Choose a value of k between 0.025 and 0.095.
2. In a lake, the value of k has been determined to be the value that you chose above, which
means that 100k% of the surface light is absorbed...
...Exponential and Logarithmic Functions
* Verify that the natural logarithm function defined as an integral has the same properties as the natural logarithm function earlier defined as the inverse of the natural exponential function.
Integrals of Exponential and Logarithmic Functions
Function  Integral 
lnx  x ∙ lnx  x + c 
logx  (x ∙ lnx  x) / ln(10) + c 
logax  x(logax  logae) + c 
ex  ex+c 
ek∙x  1 / k ∙ ek∙x + c 
ax  ax / lna + c 
xn  1 / (n+1) ∙ xn+1 + c, where n≠ 1 
1/x = x1  lnx+c 
√x = x1/2  2/3 ∙ (√x)3 + c = 2/3 ∙ x3/2 + c, where c is a constant 
Example 1: Solve integral of exponential function ∫ex32x3dx
Solution:
Step 1: the given function is ∫ex^33x2dx
Step 2: Let u = x3 and du = 3x2dx
Step 3: Now we have: ∫ex^33x2dx= ∫eudu
Step 4: According to the properties listed above: ∫exdx = ex+c, therefore ∫eudu = eu + c
Step 5: Since u = x3 we now have ∫eudu = ∫ex3dx = ex^3 + c
So the answer is ex^3 + c
Example 2: Integrate .
Solution: First, split the function into two parts, so that we get:
Trigonometric Identities and Ratio and Proportion
Trigonometric Identities
In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x =x" or usefully true, such as the Pythagorean Theorem's "a2 + b2 = c2" for right triangles. There are loads of trigonometric...
...LOGARITHMIC AND EXPONENTIAL FUNCTIONS
Inverse relations
Exponential functions
Exponential and logarithmicequations
One logarithm
THE LOGARITHMIC FUNCTION WITH BASE b is the function
y = logb x.
b is normally a number greater than 1 (although it need only be greater than 0 and not equal to 1). The function is defined for all x > 0. Here is its graph for any base b.
Note the following:
• For any base, the xintercept is 1. Why?
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
The logarithm of 1 is 0. y = logb1 = 0.
• The graph passes through the point (b, 1). Why?
The logarithm of the base is 1. logbb = 1.
•  The graph is below the xaxis  the logarithm is negative  for 

 0 < x < 1. 

 Which numbers are those that have negative logarithms? 
Proper fractions.
•  The function is defined only for positive values of x. 

 logb(−4), for example, makes no sense. Since b is always positive, no power of b can produce a negative number. 
• The range of the function is all real numbers.
• The negative yaxis is a vertical asymptote (Topic 18).
Example 1. Translation of axes. Here is the graph of the natural logarithm, y = ln x (Topic 20).
And here is the graph of y = ln (x − 2)  which is its translation 2 units to...
...Alg2  CH7 Practice Test
Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Tell whether the function y = 2( 5 ) shows growth or decay. Then graph the function. a. This is an exponential growth function. c. This is an exponential decay function.
x
b. This is an exponential growth function. d. This is an exponential growth function.
____
2. Graph the inverse of the relation. Identify the domain and range of the inverse. x y
−1
4
1 2
3 1
5 0
7 1
a.
c.
Domain: {x  0 ≤ x ≤ 4}; Range: {y  − 1 ≤ y ≤ 7} b. d.
Domain: {x  − 1 ≤ x ≤ 7}; Range: {y  − 4 ≤ y ≤ 1}
____
Domain: {x  − 7 ≤ x ≤ 1}; Domain: {x  − 7 ≤ x ≤ 1}; Range: {y  0 ≤ y ≤ 4} Range: {y  − 4 ≤ y ≤ 0} 3. Manny is a plumber and charges $50 when he visits a client and $30 per hour for every hour he works. His bill can be expressed as a function of hours, x, with the function f(x) = 50 + 30x. Which statement explains the meaning of the inverse of the function? a. Total bill as a function of the number of hours b. Cost per hour as a function of the total bill c. Number of hours as a function of the total bill d. Total bill as a function of the cost per hour
3 4. Write the exponentialequation 2 = 8 in logarithmic form. a. log38 = 2 c. log23 = 8 b. log28 = 3 d. log82 = 3
____
____
5. Evaluate log4 1 by using mental math.
16
____
a. 1 c. –2 b. 4 d. 2...
...
Equations and ProblemSolving
* An airplane accelerates down a runway at 3.20 m/s2 for 32.8 s until is finally lifts off the ground. Determine the distance travelled before takeoff.

Solutions
Given: a = +3.2 m/s2  t = 32.8 s  vi = 0 m/s 
 Find:d = ?? 
d = VI*t + 0.5*a*t2
d = (0 m/s)*(32.8 s) + 0.5*(3.20 m/s2)*(32.8 s)2
d = 1720 m

Equations and ProblemSolving
* A car starts from rest and accelerates uniformly over a time of 5.21 seconds for a distance of 110 m. Determine the acceleration of the car.

Solutions
Given: d = 110 m  t = 5.21 s  vi = 0 m/s 
 Find:a =?? 
d = VI*t + 0.5*a*t2
110 m = (0 m/s)*(5.21 s) + 0.5*(a)*(5.21 s)2
110 m = (13.57 s2)*a
a = (110 m)/ (13.57 s2)
a = 8.10 m/ s2

Equations and ProblemSolving
* Rocketpowered sleds are used to test the human response to acceleration. If a rocketpowered sled is accelerated to a speed of 444 m/s in 1.8 seconds, then what is the acceleration and what is the distance that the sled travels?

Solutions
Given: vi = 0 m/s  vf = 44 m/s  t = 1.80 s 
 Find:a =??d =?? 
a = (Delta v)/t
a = (444 m/s...
...In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative.[1][2] The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. The function is often written as exp(x), especially when it is impractical to write the independent variable as a superscript. The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.
Exponential function
Representation e^x \,
Inverse \ln x \,
Derivative e^x \,
Indefinite Integral e^x + C \,
The graph of y = ex is upwardsloping, and increases faster as x increases. The graph always lies above the xaxis but can get arbitrarily close to it for negative x; thus, the xaxis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y coordinate at that point. The inverse function is the natural logarithm ln(x); because of this, some old texts[3] refer to the exponential function as the antilogarithm.
Sometimes the term exponential function is used more generally for functions of the form cbx, where the base b is any positive real number, not necessarily e. See exponential growth for this usage.
In...