Logarithmic functions

Convert between index and logarithmic form, apply log laws and the change-of-base formula, and solve exponential equations using logarithms.

Worked examples

Converting between index and logarithmic form

Straightforward

Problem

Write

4^3 = 64

in logarithmic form. Hence evaluate

\log_4 64

and

\log_4 1

Applying log laws to simplify an expression

Moderate

Problem

Simplify

\log_2 48 - \log_2 3 + \log_2 4

without a calculator.

Solving an exponential equation using logarithms

Challenging

Problem

Solve

4^x = 100

, giving the answer correct to 2 decimal places. Then describe the relationship between

y = \log_4 x

and

y = 4^x

Practise

Q1·Straightforward

The index statement

2^5 = 32

can be written in logarithmic form as

\log_2 32 = {\square}

. What is the value of

\log_2 32

Q2·Straightforward

Evaluate

\log_3 27

Q3·Straightforward

Evaluate

\log_9 9

Q4·Straightforward

Evaluate

\log_6 1

Q5·Moderate

Use the product law

\log_a(mn) = \log_a m + \log_a n

to simplify

\log_2 16 + \log_2 4

. Give a single integer answer.

Q6·Moderate

Use the quotient law

\log_a\!\left(\dfrac{m}{n}\right) = \log_a m - \log_a n

to simplify

\log_3 81 - \log_3 3

. Give a single integer answer.

Q7·Moderate

Use the power law

\log_a(m^n) = n\log_a m

to simplify

\log_2(4^3)

. Give a single integer answer.

Q8·Moderate

Use the change-of-base formula

\log_a x = \dfrac{\log_b x}{\log_b a}

to evaluate

\log_9 27

. Write your answer as a fraction in simplest form.

Q9·Challenging

Solve

5^x = 80

, giving your answer correct to 2 decimal places.

Q10·Challenging

Solve

3^{x-1} = 50

, giving your answer correct to 2 decimal places.

Q11·Challenging

The graphs of

y = \log_3 x

and

y = 3^x

are related by a geometric transformation. Which statement correctly describes this relationship?

They are reflections of each other in the x-axis.They are reflections of each other in the y-axis.They are reflections of each other in the line $y = x$.One is a translation of the other by 1 unit.

Q12·Challenging

The graph of

y = \log_4 x

has exactly one x-intercept. Find the coordinates of this x-intercept.