Indices and logs

Question 1

What are the multiplication and division laws of logs?

Answer 1

Question 2

What is the exponent law of logs and how do you change the base of a log?

Answer 2

Question 3

What are the 2 special cases in log laws?

Answer 3

Question 4

What is an exponential function?

Answer 4

Any function of form y = a^x

Question 5

Why is n^x a reflection of (1/n)^x in the y axis?

Answer 5

Because (1/n)^x = n^-x so they are equal and opposite

Question 6

Why is euler’s number used as a natural logarithm?

Answer 6

Since any exponential function can be expressed in terms of any other exponential function, e is the most convenient base to use for any exponent or logarithm because the gradient function of e^x is the same as the graph e^x

Question 7

What do the graphs of e^x and ln x look like?

Answer 7

Question 8

What are the two forms that an exponential model can take?

Answer 8

F(t) = Ab^t
F(t) = Ae^kt
Where t represents time and the other letters are constants:
- A is initial value
- b is the growth factor for time interval of 1 (b<1 or k<0 is exponential decay, b>1 or k>0 is exponential growth)

Question 9

What does e^(ln x) and ln e^x do?

Answer 9

They are inverses (reflections in line y = x) so ln and e cancel to leace you with x.

Question 10

What are the two equation forms that can be plotted on a logarithmic scale to give a linear relationship?

Answer 10

• y = Ab^x
• y = Ax^n
  These can have logs taken from both sides to give a linear relationship on a graph.

Question 11

How do you find a linear relationship from equation of form y = Ab^x?

Answer 11

If y = Ab^x:
log y = log Ab^x
log y = log A + log b^x
log y = log A + x log b
y = c + x m
This would give a linear relationship plotted on a graph with log y on the y axis and just x on the x axis

Question 12

How do you find a linear relationship from equation of form y = Ax^n?

Answer 12

If y = Ax^n :
log y = log Ax^n
log y = log A + log x^n
log y = log A + n log x
y = c + m x
Giving a linear relationship on a graph with log y on the y axis and log x on the x axis

Question 13

What are the two equation forms that can be modelled by an exponential model?

Answer 13

• f(t) = Ab^t
• f(t) = Ae^kt
  Where t is time, A is the starting value and b or e^k are growth factors
  If b > 1 or e^k > 0; model grows exponentially.
  If b < 1 or e^k < 0; model decays exponentially.

Question 14

What does these mean (expanded)?
- (ab)^n
- a^m a^n
- a^m/a^n
- (a^m)^n

Answer 14

• a^n b^n
• a^m+n
• a^m-n
• (a^m)^n

Question 15

What do the following mean (simplified)?
- a^0
- a^-n
- a^1/n
- a^m/n

Answer 15

• 1
• 1/a^n
• nth.rt. Of a
• (nth.rt of a)^m OR nth.rt a^m

Question 16

How can you solve equations with indices without using logs?

Answer 16

Make all the bases the same (eg turn 3^x = 9 into = 3^2) and simply solve the equation of the indexes.

Question 17

What question does the value of logab answer?

Answer 17

What power of a do we need to get b?

Question 18

What are the components of log a b = x?

Answer 18

a is the base
x is the power
b is the answer

So is the same as a^x = b