Exponential and Logarithmic Functions

Question 1

What is the difference from an exponential function to a normal function?

Answer 1

• In an exponential function, the variable is in the exponent, like in F(x)=b**x.
• In a normal function, the exponent is the base.

Question 2

What is the domain and the range of any exponential function?

Answer 2

Domain: (–∞, ∞)
Range: (0, ∞)

Question 3

1) Solve 2**–3

2) Why is it so?

Answer 3

1) 1/8
2)
Take the reciprocal of the base, so 1/2, and get -3 as 3 down as the power to 2, so it gets to the denominator, so …

1           1   \_\_\_    = \_\_   2**3        8

Question 4

Where does the parent graph of any exponential function crosses the y-axis and why?

Answer 4

At (0, 1) because anything raised to the 0 power is always 1.

Question 5

Name the basic function to shift and transform an exponential graph?

Answer 5

y=a*base**(x-h) + v

a is the vertical transformation
h is the horizontal shift
v is the vertical shift

Question 6

In which relation stands a logarithm to an exponential function?

Answer 6

The logarithm (or log) is the inverse of an exponential function.

Question 7

1) How is a logarithm basically written?
2) Write 4**2 as a log!
3) What is the solution to log3 9?

Answer 7

1) logb y = x # b is leicht runter versetzt

b is base of the log,
y is the number you’re taking the log of
x is the logarithm

2)
log4 16=2

2 is called the logarithm of 16 with base 4

3) if the log is equal to nothing, set it equal to X. Then, put the base on the other side and take the x as its exponent.
log 3 9 = x
3**x = 9
Thus, x=2

Question 8

1) Make a log out of b**x = y
2) What is an Antilogarithm, also called inverse logarithm, good for?
3) What is the formula?
4) What is the solution of log2 (1/8)

Answer 8

1) logb y = x
2) If you have the logarithm and want to find out what the actual number was.
3) logb m =n -> m=b**n

4) log2 (1/8)=x
1/8 = 2x
1/23 = 2x # look if you can write exponential
2-3 = 2**x # get x up, 1 down, so denom away
-3=x

Question 9

1) What is a common logarithm?
2) What is a natural logarithm?
3) What is ln 1?

Answer 9

1) If no base is written, it is meant as base 10. Thus, log y (without a base written) is meant as log base 10.
2) A logarithm with base e (roughly equal to 2.718). The symbol for a natural log is ln. The base e isn´t written. So ln x means lne x.
3) Is actually lne 1, rewritten as lne 1 = x, thus: e**x = 1, now what is 1 if I have to put in an exponent? 0! Thus, ln1 = 0

Question 10

What is domain and range of any logarithm parent function and why?

Answer 10

Domain (0, ∞) and range (–∞, ∞).
Because an exponential parent function has the domain (–∞, ∞) and range (0, ∞). Because the log is an inverse of the exponential, domain and range switch places.

Question 11

Product rule

1) Fulfill logb x + logb y =?
2) Give an example, where both have the same base!

Answer 11

1) logb (xy)

2) log4 10 + log4 2 = log4 20.

Question 12

Quotient rule

1) Fulfill logb x - logb y =?

Answer 12

logb (x/y)

Question 13

Power rule

1) Fulfill logb x**y =?

Answer 13

1) logb x**y = y · logb x

Question 14

Change of base formula

1) What is it good for?
2) How do you proceed?

Answer 14

1) To change the base to either base 10 or 
base e (your preference) to use the buttons that your calculator does have.

2) logm n= logb n / logb m

# m and n are real numbers
# Make the new base anything you want (10 or e)

Question 15

Name the steps to graph a logarithmic parent function!

Answer 15

1. Change the log to an exponential.
   - rewrite the equation as y = log x
2. Find the inverse function by switching x and y.
   - p.e. 10**x = y.
3. Graph the inverse function.
   - plug a few x values to find y values and get points
4. Reflect every point on the inverse function graph over the line y = x.

Question 16

How to graph a transformed log?

Answer 16

1) Find the parent function logb x.
2) Do the same steps as with logarithmic parent function.
3) Then do the shifts.

Remember! Transformed logs are f(x) = a · logb(x – h) + v (a is vertical stretch/shrink, h is horizontal shift, v is vertical shift)

Question 17

Solving logarithm equations. There are 4 types:

1) Variable is inside the log: log3 x = –4
2) Variable is the base: logx 16 = 2
3) Variable is inside the log, but the equation has more than one log and a constant: log2(x – 1) + log2 3 = 5
4) Variable is inside the log, and all terms in the equation involve logs: log3(x – 1) – log3(x + 4) = log3 5

ALWAYS put the solution back in the original formula. There are no negative numbers allowed in a log, so if last line is ln (−10)+ln(−7)=ln(70) # NOT OKAY. Befor you have to plug in and log what parentheses give, p.e. (x+10)

Answer 17

1) Change to an exponential equation: 1/81=X
2) Change to an exponential equation: x² = 16, so X equals ±4. But logs don’t have negative bases, so X=4

3) Combine the two logs by using the product rule: log2[(x – 1) · 3] = 5. Turn this into 2⁵ = (x – 1) · 3
Solution: X=35/3

4. • They need same base to solve.
   • Use quotient rule to get log3 x-1/x+4 = log3 5.
   • Drop log base 3 from both sides.
   • Solve and plug result into part of the initial equation, p.e. log3(x – 1).
   • Because you cant have a negative number inside the log, there isno solution

Question 18

Name the formula for exponential word problems and what the variables stand for!

Answer 18

B(t) = Pe**rt #both r and t are squared

B(t)= value of how many people, bacteria... you have after time t. 
P = number of objects when t = 0. 
t = time 
r = constant that describes rate at which population is changing. If r is positive = growth constant. If r is negative =decay constant. 
e= base of the natural logarithm, used for continuous growth or decay.