Solving Exponential and Logarithmic Equations
Exponential Equations (variable in exponent position)
1. Isolate the exponential portion ( base exp onent ): Move all non-exponential 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 non-log 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...

...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...

...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...

...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 |...

...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...

...APPLICATIONS OF EXPONENTIAL|
AND|
LOGARITHMIC FUNCTIONS|
EARTHQUAKE WORD PROBLEMS:
As with any word problem, the trick is convert a narrative statement or question to a mathematical statement.
Before we start, let's talk about earthquakes and how we measure their intensity.
In 1935 Charles Richter defined the magnitude of an earthquake to be
where I is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken...

...-------------------------------------------------
Equations and Problem-Solving
* 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 take-off.
-------------------------------------------------
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...

...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...

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