Solving Exponential Equations using Logarithms

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Solving Exponential Equations using Logarithms — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionApplying logarithms to both sidesAssessment Practice

5 marks~8 minCriterion A

The diagram shows part of the graph of

f(x) = \log_5 x

for

x > 0

Write down the value of

f(125)

. [1]

The graph of

g

is obtained by reflecting

f

in the

y

-axis and then translating it by

\begin{pmatrix}3\\0\end{pmatrix}

. Find an expression for

g(x)

. [2]

Deduce the equation of the vertical asymptote of

g

and hence state the domain of

g

. [2]

Solutions

2QuestionApplying logarithms to both sidesAssessment Practice

6 marks~9 minCriterion C

The table below shows four exponential equations of the form

a^x = a^2 + 1

.

Base

a

: 2, 3, 4, 5

Equation:

2^x = 5

3^x = 10

4^x = 17

5^x = 26

Approximate solution

x

2.322

2.096

2.044

2.024

Write down the exact solution of

3^x = 10

in the form

x = \dfrac{\log p}{\log q}

, where

p

and

q

are integers. [1]

Find the exact solution of

a^x = a^2 + 1

in terms of

a

, and hence show that

x > 2

for all valid bases

a

, and that

x \to 2

a \to \infty

. [3]

A student claims: "For a very large base

a

, the equation

a^x = a^2 + 1

is approximately the same as

a^x = a^2

, so

x \approx 2

."

Justify whether the student's reasoning is correct, and state one limitation of using

x = 2

as an approximation for small values of

a

. [2]

Solutions

3QuestionIsolating exponential expressionsAssessment Practice

9 marks~14 minCriterion B

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

n

consists of a column of

n

squares on the left and a row of

n

squares along the bottom, sharing one corner square.

Figure ( $n$ )	1	2	3	4
Squares ( $S$ )	1	3	5	7

Find the number of squares in Figure 6 and Figure 10. [2]

Show that the number of squares in Figure

n

is given by

S(n) = 2n - 1

. Verify your rule for Figure 4. [3]

A second pattern is created where each Figure

n

is formed by placing two copies of the original L-shape together, sharing their corner squares. The number of squares in Figure

n

of the second pattern is

T(n)

.

Show that

T(n) = 4n - 3

, and hence justify whether

n = 26

is the only positive integer solution to

T(n) = S(n) + 50

. [4]

Solutions

4QuestionApplying logarithms to both sidesAssessment Practice

11 marks~17 minCriterion D

The table below shows four data points believed to follow a relationship of the form

y = A \cdot 3^x + B

$x$	1	2	3	4
$y$	7	19	55	163

Show that the first differences of

y

have a common ratio of 3, and hence find the values of

A

and

B

by solving a pair of simultaneous equations. [4]

Show that plotting

\log_{10}(y - B)

against

x

produces a linear relationship with gradient

\log_{10} 3

, and verify this using at least two data points from the table. [3]

Justify why

\log_{10}

cannot be applied directly to both sides of

y = A \cdot 3^x + B

to linearise the model, and assess whether the model

y = 2 \cdot 3^x + 1

provides a reliable basis for predicting values beyond

x = 4

. [4]

Solutions

Questions 5-113

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