### Question

(i) Show that the equation $$log_{10}(x-4)=2-log_{10}x$$ can be written as a quadratic equation in x.

(ii) Hence solve the equation    $$log_{10}(x-4)=2-log_{10}$$ x giving your answer correct to 3 significant figures.

1(i) Use law for the logarithm of a product or quotient

$$Use log_{10}100=2 or 10^{2}=100$$
obtain$$x^{2}-4x-100=0$$ , or equivalent

1(ii) Solve a 3-term quadratic equation

### Question

Showing all necessary working, solve the equation ln(2x − 3 )= 2 ln x − ln (x − 1). Give your answer correct to 2 decimal places.

Use law for the logarithm of a product ,quotient or power

Obtain a correct equation free of logarithms

Solve a 3-term quadratic obtaining at least one root

### Question

It is given that 2 ln(4x − 5 )+ ln(x + 1 )3 ln 3.(i) Show that $$16x^{3}-24x^{2}-15x-2=0$$
(ii) By first using the factor theorem, factorise $$16x^{3}-24x^{2}-15x-2=0$$ completely.

(iii) Hence solve the equation 2 ln (4x − 5 )+ ln(x + 1) = 3 ln 3.

(i) Use law for the logarithm for a product or quotient or exponentiation AND for a power

Obtain$$(4x-5)^{2}(x+1)=27$$

Obtain given equation correctly $$16x^{3}-24x^{2}-15x-2=0$$

(ii) Obtain x = 2 is root or (x – 2) is a factor, or likewise with$$x=-\frac{1}{4}$$

Divide by (x – 2) to reach a quotient of the form$$(16x^{3}+kx)$$

Obtain quotient $$16x^{3}+8x+1$$

(iii) State x = 2 only

### Question

(i) Show that the equation

$$log_{10}$$ (x-4)=2-$$log_{10}x$$can be written as a quadratic equation in x.

(ii) Hence solve the equation giving your answer correct to 3 significant figures.

(i) Use law for the logarithm of a product or quotient Use $$log_{10}100=2or 10^{2}=100$$Obtain $$x^{2}-4x-100=0$$,or equivalent

### Question

Two variable quantities x and y are believed to satisfy an equation of the form $$y=C(a^{x})$$, where C and a are constants. An experiment produced four pairs of values of x and y. The table below gives the corresponding values of x and ln y. By plotting ln y against x for these four pairs of values and drawing a suitable straight line, estimate the values of C and a. Give your answers correct to 2 significant figures. 2 Plot the four points and draw straight line
State or imply that ln y=ln C+x lna
Carry out a completely correct method for finding lnC or ln a

### Question

It is given that x = ln (1 − y) − ln y, where 0 < y < 1.

(i) Show that $$y=\frac{e^{-x}}{1+e^{-x}}$$.

(ii) Remove logarithms correctly and obtain  $$e^{x}=\frac{1-y}{y}$$

(ii) Hence show that $$\int_{0}^{1}ydx=ln(\frac{2e}{e+1})$$

Obtain the given answer  $$y=\frac{e^{-x}}{1+e^{x}}$$ following full working

3(ii) State integral k ln $$(1+e^{-x})$$where k = ± 1
State correct integral- ln $$(1+e^{-x})$$
Use limits correctly D
$$ln\left ( \frac{2e}{e+1} \right )$$  following full working

### Question

Solve the equation $$(1+2^{x})=2$$, giving your answer correct to 3 decimal places.

Remove logarithm and obtain $$1+2^{x}=e^{2}$$

Use correct method to solve an equation of the form $$2^{x}$$ = a , where a > 0Obtain answer x = 2.676

### Question

Solve the equation

log10 (2x + 1) = 2 log10 (x + 1) − 1

State or imply 10 log 10 1 = Use law of the logarithm of a power, product or quotient
Obtain a correct equation in any form, free of logs
Reduce to x2 − 18x − 9 =  0, or equivalent
Obtain final answers x = 18.487 and x = –0.487

### Question

Find the set of values of x for which $$2(3^{1-2x})< 5^{x}$$. Give your answer in a simplified exact form. 

Ans

1  Use law of the logarithm of a product or power

Obtain a correct linear inequality in any form, e.g. ln 2+ (1–2x) ln 3 <x ln 5

Solve for x

Obtain $$x> \frac{ln 6}{ln 45}$$

### Question

Solve the equation In 3 + In(2x + 5) = 2 In(2x + 5) = 2 In (x + 2). Give your answer in a simplified exact form.

Ans:

Use law of logarithm of a power and sum and remove logarithms
Obtain a correct equation in any form, e.g. $$3(2x+5)=(x+2)^2$$
Use correct method to solve a 3-term quadratic, obtain at least one root
Obtain final answer final answer $$x=1+2\sqrt{3}$$ or $$1+\sqrt{12}$$ only

### Question

Find the real root of the equation $$\frac{2e^{x}+e^{-x}}{2+e^{x}}=3$$,  giving your answer correct to 3 decimal places.
Your working should show clearly that the equation has only one real root.                                                                                       

Ans

2 Reduce to a 3-term quadratic u2 + 6u −1= 0 OE

Solve a 3-term quadratic for u

Obtain root  $$\sqrt{10}-3$$

Obtain answer x = – 1.818 only

Reject  $$-\sqrt{10}-3$$ correctly

Alternative method for Question 2

Rearrange to obtain a correct iterative formula

Use the iterative process at least twice

Obtain answer x = – 1.818

Show sufficient iterations to at least 4 d.p. to justify x = – 1.818

Clear explanation of why there is only one real root

### Question

Solve the equation $$In(x^3-3)=Inx-In3$$. Give your answer correct to 3 significant figures.

Ans:

Use law of the logarithm of a product or power
Obtain a correct equation free of logarithms, e.g. $$3(x^3-3)=x^3$$
Obtain x = 1.65

### Question

Solve the equation $$(1+2^{x})=2$$, giving your answer correct to 3 decimal places.

Remove logarithm and obtain $$1+2^{x}=e^{2}$$

Use correct method to solve an equation of the form $$2^{x}$$ = a , where a > 0

### Question

Solve the equation  = 8, giving your answer correct to 3 decimal places.

Solve for$$3^{x}$$and obtain$$3^{x}=\frac{18}{7}$$

Use correct method for solving an equation of the form

Obtain answer x = 0.860 3 d.p. only

### Question

Solve the equation ln(x2 + 4) = 2 ln x + ln 4, giving your answer in an exact form. 

Ans:

Use law of the logarithm of a power, quotient or product
Remove logarithms and obtain a correct equation in x, e.g. x2 + 4 = 4x2
Obtain final answer x = 2/ √3 , or exact equivalent

### Question The variables x and y satisfy the equation xny2 = C, where n and C are constants. The graph of ln y against ln x is a straight line passing through the points (0.31, 1.21) and (1.06, 0.91), as shown in the diagram.
Find the value of n and find the value of C correct to 2 decimal places.

Ans:

State or imply n In x + 2 In y = In C

Substitute values of ln y and ln x, or equate gradient of line to $$\pm \frac{1}{2}n$$  but not ±n, and solve for n

Obtain n = 0.8 or 0.8 or  $$\frac{4}{5}$$

Solve for C

Obtain C = 14.41

### Question

Solve the equation

$$5^{x−1}$$=$$5^{x }− 5$$,

EITHER Use laws of indices correctly and solve for 5^{x} or for $$5^{–x}$$
or for $$5^{x–1}$$
Obtain$$5^{x}$$
or for$$5^{–x}$$
or for$$5 ^{x–1}$$
in any correct form, e.g. $$5^{x}$$$$\frac{5}{1-\frac{1}{5}}$$

Use correct method for solving $$5^{x}$$ = a, or $$5^{–x}$$ = a, or $$5^{x–1}$$ = a, where a 0  Obtain answer x = 1.14

OR Use an appropriate iterative formula, $$x_{n+1}$$=$$\frac{In(5^{x-1+5})} {In5}$$correctly, at least once

Show sufficient iterations to at least 3 d.p. to justify 1.14 to 2 d.p., or show there is a sign change in the interval (1.135, 1.145)  Show there is no other root  [For the solution x = 1.14 with no relevant working give , and a further B1 if
1.14 is shown to be the only solution.]

### Question

The polynomial p(x) is defined by$$p(x)=x^{3}-3ax+4a$$,where a is a constant.
(i) Given that (x − 2) is a factor of p(x), find the value of a.
(ii) When a has this value,
(a) factorise p(x)completely,
(b) find all the roots of the equation$$p(x^{2})$$=0

(i) Substitute x = 2 and equate to zero, or divide by x – 2 and equate constant remainder to zero, or equivalent
Obtain a = 4

(ii) (a) Find further (quadratic or linear) factor by division, inspection or factor theorem or equivalent

Obtain x^{2}+ 2x – 8 or x + 4

State (x – 2)^{2}

(x + 4) or equivalent

(b) State any two of the four (or six) roots
State all roots $$(\pm \sqrt{2},\pm 2i)$$provided two are purely imaginary

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