Graphs of Logarithmic Functions

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Graphs of Logarithmic Functions — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionBasic log function graphsConcept Practice

2 marks~3 minCriterion A

The diagram shows the graph of

y = \log_2 x

State the domain of

y = \log_2 x

. [1]

State the range of

y = \log_2 x

. [1]

Solutions

2QuestionComparing exponential and logarithmic graphsAssessment Practice

5 marks~8 minCriterion C

The diagram shows the graphs of

y = 2^x

and

y = \log_2 x

for

-3 \leq x \leq 5

Write down the equation of the asymptote of

y = \log_2 x

. [1]

Both

y = 2^x

and

y = \log_2 x

intersect the line

y = x

. Deduce the geometric relationship between the graphs of

y = 2^x

and

y = \log_2 x

. [2]

A student claims that

2^x > \log_2 x

for all

x > 0

. Justify whether this claim is correct. [2]

Solutions

3QuestionComparing exponential and logarithmic graphsAssessment Practice

6 marks~9 minCriterion D

A biologist records the size of a bacterial colony at hourly intervals. The observed counts (in thousands of bacteria) are shown below.

$x$ (hours)	1	2	4	8
Observed $B$ (thousands)	2	4	8	14

The biologist proposes the exponential model

B(x) = 2^x

, where

B

is the population in thousands and

x

is the time in hours. The diagram shows the curve

B(x) = 2^x

and the four observed data points.

Complete the table of predicted values using

B(x) = 2^x

$x$ (hours)	1	2	4	8
Predicted $B$ (thousands)	2	4	\_\_\_	\_\_\_ [1]

Calculate the percentage error between the predicted and observed values at

x = 4

and at

x = 8

, giving your answers to 1 decimal place.

\text{percentage error} = \frac{|\text{predicted} - \text{observed}|}{\text{observed}} \times 100\%

[2]

Using your results from part (b), justify whether the biologist should continue using the exponential model

B(x) = 2^x

to predict bacterial population beyond

x = 8

hours, and identify one biological reason why exponential growth cannot continue indefinitely. [3]

Solutions

4QuestionComparing exponential and logarithmic graphsAssessment Practice

8 marks~12 minCriterion B

The diagram shows the graphs of

y = 2^x

and

y = \log_2 x

drawn on the same axes for

x > 0

.

The table below records selected values of both functions.

$x$	1	2	4	8
$y = 2^x$	2	4	16	256
$y = \log_2 x$	0	1	2	3

Describe the pattern in the values of

y = \log_2 x

x

doubles from 1 to 2, from 2 to 4, and from 4 to 8. Hence deduce the value of

\log_2 32

. [2]

A student claims that when

x

doubles, the value of

y = 2^x

is squared. Using values from the table, verify this claim for two consecutive doublings of

x

, and find the value of

n

such that

2^n = 256^2

. [3]

The two graphs are reflections of each other in the line

y = x

. Using the identity

\log_2(2^x) = x

and a pair of corresponding table values, justify whether

y = \log_2 x

and

y = 2^x

are inverse functions, and state the domain and range of

y = \log_2 x

. [3]

Solutions

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