Exponential–Logarithmic Relationships

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Exponential–Logarithmic Relationships — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionConverting exponential equations to logarithmic formAssessment Practice

5 marks~8 minCriterion A

The diagram shows part of the graph of

f(x) = \log_{10} x

for

x > 0

Write down the value of

f(1000)

. [1]

Find the value of

x

such that

f(x) = f(25) + f(4)

, giving your answer as an exact integer. [2]

Given that

f(x^2) - f(x) = 2

, deduce the value of

x

and explain what this result reveals about the relationship between parts (b) and (c). [2]

Solutions

2QuestionConverting exponential equations to logarithmic formAssessment Practice

7 marks~11 minCriterion B

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

Figure 1 has 3 squares, Figure 2 has 8 squares, and Figure 3 has 15 squares.

Write down the number of 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

, where

a

and

b

are integers. [3]

Justify why the rule must contain an

n^2

term by referring to the structure of the figures, and verify that your rule gives the correct count for Figure 3. [3]

Solutions

3QuestionConverting exponential equations to logarithmic formAssessment Practice

8 marks~12 minCriterion D

A savings account pays compound interest at a fixed annual rate. After

t

years, the balance

B

(in dollars) is modelled by

B(t) = 1000 \cdot r^{\,t}

where

r

is the growth factor. The account balance after 2 years is 1210 dollars.

Show that

r = 1.1

. [2]

Find the number of complete years it takes for the balance to first exceed 1600 dollars. Give a clear algebraic method using logarithms. [3]

A second account compounds interest every 6 months at half the annual rate, giving actual balances that differ from the original model. The percentage errors between the two models are:

t

(years): 2, 5, 10

Percentage error (\%): 0.46, 1.12, 2.25

Assess whether a long-term investor should rely on the original model to predict their balance, justifying your answer by citing the trend in the percentage errors above. [3]

Solutions

4QuestionConverting exponential equations to logarithmic formAssessment Practice

9 marks~14 minCriterion C

The diagram shows the graphs of

f(x) = 5^{x+2}

and

g(x) = 3^{2x-1}

for

-2 \le x \le 5

Show that the equation

5^{x+2} = 3^{2x-1}

can be written as

x = \frac{\log 3 + 2\log 5}{2\log 3 - \log 5}.

[3]

Hence find the value of

x

, giving your answer correct to 3 significant figures. Verify that your value satisfies

5^{x+2} = 3^{2x-1}

by evaluating both sides. [3]

A student claims that because

5^{x+2} = 3^{2x-1}

has exactly one solution, the graphs of

f

and

g

can intersect at most once for all real

x

. Justify whether this claim is correct, referring to the nature of exponential functions. [3]

Solutions

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