C2 - Logarithms

Question 1

3.1.1 What is a logarithm?

Answer 1

A logarithm gives the power that a number must be raised to in order to yield a result. In better words:

If x = a^m, then m = log(a)(x) where a is written as subscript.

Question 2

3.1.2 What is the base of a logarithm?

Answer 2

In m = log(a)(x), where a is in subscript, the base of the log is a. The base is the original number that had been raised to a power.

Question 3

3.1.3 What does it mean if the log is written with no base?

Answer 3

If a log is written with no base, such as log(10), it is automatically implied that there base of the log is 10.

Log(10)(x) == log(x)

Question 4

3.1.4 What is the relationship between index laws and log laws?

Answer 4

As logs use indices to work, it’s easy to see how log laws have been derived from the respective index laws.

Question 5

3.1.5 Log laws: adding and subtracting indices

Answer 5

if x = a^m and y = a^n, then xy = a^(m+n).

Similarly, that means m + n = log(xy) = log(x) + log(y)

It also means log(x) - log(y) = log(x/y)

Question 6

3.1.6 Log laws: powers in a log function

Answer 6

Log(x^n) = nlog(x)

Question 7

3.1.7 Log laws: definite answers from index laws

Answer 7

Log(1) = 0

Log(a)(a) = 1 when the base is equal to the result of the power statement.

Question 8

3.1.8 Reducing to linear form for graphs.

Answer 8

If y = ab^x:

Taking logs of both sides gives log(y) = Log(a) + xlog(b) in the form of y = mx + c.
Where plotting log(y) against x gives intercept of log(a) and a gradient of log(b).

If y = ax^n:

Taking logs of both sides gives log(y) = log(a) + nlog(x) in the form of y = mx + c.
Where plotting log(y) against log(x) gives gradient n and intercept log(a).