Solve the inequality 3 × 2 3 x + 2 < 8 , giving your answer in terms of logarithms.

Given Inequality 3 × 2 3 x + 2 < 8 ⇒ 2 3 x + 2 < 8 3 ⇒ 2 3 x × 2 2 < 8 3 ⇒ 2 3 x × 4 < 8 3 ⇒ 2 3 x < 2 3 Taking natural logarithms on both sides ⇒ ln ( 2 3 x ) < ln 2 3 Use the power law of logarithms, ln u m = m ln u ⇒ ( 3 x ) ln ( 2 ) < ln 2 3 ⇒ x < ln 2 3 ( 3 ) ln ( 2 ) Use power law again ⇒ x < ln 2 3 ln ( 2 3 ) ⇒ x < ln 2 3 ln ( 8 ) ⇒ x < ln 2 3 ln ( 8 )

Given that ( 1 . 2 ) x = 6 y , use logarithms to find the value of x y correct to 3 significant figures.

Given 1 . 2 x = 6 y Taking logarithm on both sides ⇒ log 1 . 2 x = log 6 y Use the power law of logarithms, ln u m = m ln u ⇒ x log 1 . 2 = y log 6 ⇒ x y = log 6 log 1 . 2 ⇒ x y = 0 . 7781 0 . 0791 ⇒ x y = 9 . 8274 ⇒ x y = 9 . 83 (rounded to 2 decimal places).

EASY

AS and A Level

IMPORTANT

Earn 100

Given that $\ln p = 2 \ln q - \ln (3 + q)$ and that $q > 0$ , express $p$ in terms of $q$ not involving logarithms.

Important Questions on Logarithmic and Exponential Functions

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>END-OF-CHAPTER REVIEW EXERCISE 2>Q 3

Solve the inequality

3 \times 2^{3 x + 2} < 8

, giving your answer in terms of logarithms.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>END-OF-CHAPTER REVIEW EXERCISE 2>Q 4

Use logarithms to solve the equation
$5^{x + 3} = 7^{x - 1},$

giving the answer correct to $3$ significant figures.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>END-OF-CHAPTER REVIEW EXERCISE 2>Q 5

Solve the equation

6 (4^{x}) - 11 (2^{x}) + 4 = 0

, giving your answers for

x

in terms of logarithms where appropriate.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>END-OF-CHAPTER REVIEW EXERCISE 2>Q 6

Solve the equation

\ln (5 x + 4) = 2 \ln x + \ln 6 .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>END-OF-CHAPTER REVIEW EXERCISE 2>Q 7

Question Image

The variables $x$ and $y$ satisfy the equation $y = K x^{m}$ , where $K$ and $m$ are constants. The graph of $\ln y$ against $\ln x$ is a straight line passing through the points $(0, 2.0)$ and $(6, 10.2)$ , as shown in the diagram. Find the values of $K$ and $m$ , correct to $2$ decimal places.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>END-OF-CHAPTER REVIEW EXERCISE 2>Q 8

Given that $y = 2^{x}$ , show that the equation $2^{x} + 3 (2^{- x}) = 4$ can be written in the form $y^{2} - 4 y + 3 = 0 .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>END-OF-CHAPTER REVIEW EXERCISE 2>Q 8

Hence Solve the equation $2^{x} + 3 (2^{- x}) = 4,$ giving the values of $x$ correct to $3$ significant figures where appropriate.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 2 - Logarithmic and Exponential Functions>END-OF-CHAPTER REVIEW EXERCISE 2>Q 9

Given that

(1.2)^{x} = 6^{y}

, use logarithms to find the value of

\frac{x}{y}

correct to

3

significant figures.

Given that lnp=2lnq-ln(3+q) and that q>0, express p in terms of q not involving logarithms.

Important Questions on Logarithmic and Exponential Functions

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Given that $\ln p = 2 \ln q - \ln (3 + q)$ and that $q > 0$ , express $p$ in terms of $q$ not involving logarithms.