Introduction to Logarithms

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Introduction to Logarithms — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionCommon logarithms (base 10)Assessment Practice

5 marks~8 minCriterion B

The diagram shows the first four figures of a pattern made from small squares arranged in an L-shape.

Write down the number of small squares in Figure 4. [1]

Find a rule for the total number of squares

S

in Figure

n

, giving your answer in the form

S = an^2 + bn + c

, where

a

b

and

c

are integers to be found. [2]

Verify your rule for Figure 3. Hence justify why the rule must contain an

n^2

term by referring to the structure of the L-shape. [2]

Solutions

2QuestionCommon logarithms (base 10)Assessment Practice

5 marks~8 minCriterion C

The diagram shows part of the graph of

f(x) = \log_{10}(x)

Write down the value of

f(100)

. [1]

Given that

f(x) = -1.5

, find the exact value of

x

. [2]

A student claims that

\log_{10}(a^2) = 2\log_{10}(a)

holds for all real values of

a

. Justify whether this claim is correct, and state the values of

a

for which it is valid. [2]

Solutions

3QuestionCommon logarithms (base 10)Assessment Practice

6 marks~9 minCriterion D

The Richter scale measures earthquake magnitude using a logarithmic model. The intensity ratio of two earthquakes with magnitudes

M_1

and

M_2

is given by

\text{Intensity ratio} = 10^{(M_1 - M_2)}.

The 1906 San Francisco earthquake had magnitude

M_1 = 7.9

. A seismologist compares predicted intensity ratios (relative to the 1906 earthquake) with observed field measurements for three minor tremors.

Tremor magnitudes: A:

3.2

— B:

4.4

— C:

5.1

Predicted ratio: A: ? — B: ? — C: ?

Observed ratio: A:

48\,000

— B:

2900

— C:

640

Calculate the predicted intensity ratio for each tremor. Give each answer to 3 significant figures. [2]

For tremor B, calculate the percentage error between the predicted and observed intensity ratios using

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

Give your answer to 1 decimal place. [2]

A journalist claims the Richter scale model is equally reliable for all three tremors. Justify whether this claim is supported by your results, and identify one limitation of using the Richter scale formula to predict intensity ratios in the field. [2]

Solutions

4QuestionCommon logarithms (base 10)Assessment Practice

8 marks~12 minCriterion A

The diagram shows part of the graph of

f(x) = \log_{10}(x)

Find the value of

a

such that

f(a) = f(25) + f(4)

, giving your answer as an integer. [2]

Solve the equation

\log_{10}(x+3) + \log_{10}(x-1) = 1

, rejecting any value of

x

that is not in the domain of

f

. [4]

Justify whether the point

(50, 2)

lies above or below the graph of

f(x) = \log_{10}(x)

. [2]

Solutions

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