Mathematics, five units · Grade 12
What is the natural logarithm and how do you differentiate it?
The natural logarithm answers the question: to what power must e be raised to give this number. It is the inverse of the exponential function, its derivative is one over x, and its domain is the positive numbers only — which is a restriction that has to be stated in every answer, not assumed.
Learning objectives
- Solve an equation in which the unknown sits inside a logarithm
- Differentiate a function containing a natural logarithm
- Investigate a function that multiplies a logarithm by a polynomial
- Explain the connection between the logarithm and the exponential through their graphs
What it says
The natural logarithm of a number is the power to which e must be raised to produce it. It is the exponential function read backwards, which is the whole of its definition.
Because e to any power is positive, the logarithm only accepts positive inputs. The domain restriction is not a technicality to mention at the end; it decides which of your solutions are real ones.
That is also why a logarithmic equation must be checked at the end. A value that solves the algebra but makes any logarithm's input zero or negative is not a solution at all.
The derivative
The natural logarithm of x differentiates to one over x. It follows from the inverse relationship with the exponential, and it is short enough to be worth simply knowing.
With an expression inside, the chain rule gives the derivative of the inside over the inside itself. That form is worth recognising, because it appears constantly in five-unit investigations.
Log laws that shorten the work
The logarithm of a product is the sum of the logarithms, of a quotient the difference, and of a power the exponent times the logarithm.
Applying these before differentiating often removes the need for the product, quotient or chain rule entirely. A logarithm of a complicated product becomes a sum of simple terms, and the differentiation that follows is trivial.
Worked examples
Differentiate the natural logarithm of 3x
- By the log laws this is the logarithm of 3 plus the logarithm of x
- The logarithm of 3 is a constant, so it differentiates to zero
- The logarithm of x differentiates to one over x
Answer: One over x
Differentiate the natural logarithm of (x squared + 1)
- There is an expression inside, so use the chain rule
- The derivative of the inside is 2x
- The result is the inner derivative over the inside
Answer: 2x over (x squared + 1)
What is the domain of the natural logarithm of (x − 4)?
- A logarithm accepts positive inputs only
- So x minus 4 must be greater than zero
- Solve the inequality
Answer: x greater than 4
Common mistakes
- Ignoring the domain
- A logarithm is defined only for positive inputs. A solution making the input zero or negative must be discarded, and forgetting this turns a correct calculation into a wrong answer.
- Treating the logarithm of a sum as a sum of logarithms
- The law applies to a product, not a sum. There is no rule for the logarithm of a sum, and inventing one is a common and costly slip.
- Dropping the inner derivative
- With an expression inside, the derivative is the inner derivative over the inside. Writing one over the inside alone omits the chain rule.
What to remember
- The logarithm is the inverse of the exponential.
- Its derivative is one over x.
- The domain is positive inputs only — always check.
- Log laws first, differentiation second.
More in Mathematics, five units
- How do you prove a trigonometric identity?
- How do you solve a trigonometric equation over a full period?
- How do you find the asymptotes of a rational function?
- How do you differentiate a root and an exponential function?
- What is a definite integral and how do you calculate it?
- How do you find the length of a vector and the angle between two vectors?
- What is the normal distribution and how do you standardise a measurement?