Worksheet: Logarithmic Function
z8. Page 2. 4. Write the following equalities in exponential form. (1) log3 81 = 4. (2) log7 7 = 1. (3)
EXPONENTIALS & LOGARITHMS
ln9 ln28 ln12 ln49 x x. +. = + giving the answer as an exact fraction. 1. 2 x = Question 14 (***). Rearrange each of the following equations for x .
Higher Mathematics EXPONENTIALS & LOGARITHMS
It is common in applications to find an exponential relationship between exponential and logarithmic functions. EXAMPLES. 1. Shown below is the graph ...
Exponential and Logarithmic Equations
Solution (a): To solve this equation we will use the guidelines for solving exponential equations given above. Step 1: The first step in solving an exponential
Logarithmic Functions
with a > 0 and a ≠ 1 is a one-to-one function by the Horizontal Line. Test and therefore has an inverse function. The inverse function of the exponential.
Chapter 6: Exponential and Logarithmic Functions Selected
Chapter 6: Exponential and Logarithmic Functions. Selected Solutions to Odd Problems. 124. Section 6.3 To answer these questions I need to know what each part.
13EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Example. 1 page 928
Exponential and Logarithmic Functions
extraneous solutions with logarithmic equations it is wise to determine the domain Chapter Test Prep Videos include step-by-step solutions to all chapter ...
Derivative of exponential and logarithmic functions
functions which involve exponentials or logarithms. Example. Differentiate loge (x2 + 3x + 1). Solution. We solve this by using the chain rule and our ...
Worksheet: Logarithmic Function
Write the following equalities in exponential form. (1) log3 81 = 4. (2) log7 7 = 1. (3) log1. 2.
LOGARITHMS EXAM QUESTIONS
Created by T. Madas. Question 26 (***+). Solve each of the following exponential equations giving the final answers correct to. 3 significant figures.
Sample Exponential and Logarithm Problems 1 Exponential Problems
Sample Exponential and Logarithm Problems Solution: Note that ... Using the power of a power property of exponential functions we can multiply the ...
Exponential and Logarithmic Functions
Chapter 10 is devoted to the study exponential and logarithmic functions. 10.7 Logarithmic and Exponential Equations ... Skill Practice Answers.
Exponential and Logarithmic Functions
The correspondence from x to f1g1x22 is called a composite function f ? g. 1. 4.1 Composite Functions. Now Work the 'Are You Prepared?' problems on page 286.
Higher Mathematics EXPONENTIALS & LOGARITHMS
4 Exponentials and Logarithms to the Base e Remember that the graph of an exponential function ( ) ... expressing your answer in the form log log.
Solving Exponential & Logarithmic Equations
Original Equation. Rewrite with like bases. Property of exponential equations. Subtract 2 from both sides. The solution is 1. Check this in the original
DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC
find second order derivative of a function. l state Rolle's Theorem and Lagrange's Mean Value Theorem; and l test the
derivative-of-exponential-and-logarithmic-functions.pdf
Differentiate loge (x2 + 3x + 1). Solution. We solve this by using the chain rule and our knowledge of the derivative of loge x. d dx.
Integrals Exponentials and Logarithms Techniques of Integration
What is dxldy when x is the logarithm logby? Thpse questions are closely related because bx and logby are inverse functions. If one slope can be found
Created by T. Madas
Created by T. Madas
LOGARITHMS
EXAMQUESTIONS
Created by T. Madas
Created by T. Madas
Question 1 (**)
Show clearly that
1log 36 log 256 2log 48 log 42a a a a+ - = -.
proofQuestion 2 (**)
Simplify
2 2log 5 log 1.6+,
giving the final answer as an integer. 3Question 3 (**+)
Given that 2px= and 4qy=, show clearly that
32log ( ) 3 2x y p q= +.
proofCreated by T. Madas
Created by T. Madas
Question 4 (**+)
Simplify each of the following expressions, giving the final answer as an integer. a) 2 2log 3 log 24-. b) 21log 4loga aaaÄ Ô-Å ÕAE Ö, 0a>, 1a≠.
Full workings, justifying every step, must support each answer. 3-, 6Question 5 (**+)
Given that 2logy x=, write each of the following expressions in terms of y. a) 22logx b) ()22log 8x2y, 3 2y+
Created by T. Madas
Created by T. Madas
Question 6 (**+)
Given that 24 10xy= × express x in terms of y, giving an exact simplified answer in terms of logarithms base 10. ()101 1log2 4x y=Question 7 (**+)
An exponential curve has equation
xy ab=, x??, where a and b are non zero constants. Make x the subject of the above equation, giving the final answer in terms of logarithms base 10. log log log y axb-=Created by T. Madas
Created by T. Madas
Question 8 (**+)
Solve the following logarithmic equation
10 10 102log log 3 log 75x+ =.
5, 5x x= ≠ -
Question 9 (**+)
Solve the following logarithmic equation
log log ( 3) log 10a a ax x+ - =.C2H, 5, 2x x= ≠ -
Created by T. Madas
Created by T. Madas
Question 10 (***)
An exponential curve C has equation
13xy=, x??.
a) Sketch the graph of C. b) Solve the equation 23y=, giving the answer correct to 3 significant figures.C2B, 0.369
Created by T. Madas
Created by T. Madas
Question 11 (***)
Given that
log 4ap= and log 5aq=, express each of the following logarithms in terms of p and q. a) log 100a b) log 0.4aThe final answers may not contain any logarithms.
C2G, 2p q+, 12p q-
Question 12 (***)
Solve the following logarithmic equation
5 5log (4 7) log 2t t+ - =.
C2L, 13t=
Created by T. Madas
Created by T. Madas
Question 13 (***)
Given that
2log 3p= and 2log 5q=,
express each of the following logarithms in terms of p and q. a) 2log 45 b) 2log 0.3The final answers may not contain any logarithms.
MP1-N, 2p q+, 1p q- -
Created by T. Madas
Created by T. Madas
Question 14 (***)
Solve each of the following equations, giving the final answers correct to three significant figures, where appropriate. a) 7 10x=. b) 2 29loglogyy=.
C2J, 1.18x≈, 1, 88y=
Question 15 (***)
Solve the following logarithmic equation for x.
2log ( 10) log 2log 3a a ax x- - =.
10, 1x x= ≠ -
Created by T. Madas
Created by T. Madas
Question 16 (***)
Solve the following logarithmic equation for x.
2log log 18 log ( 4)a a ax x= + -.
C2C, 6, 12x=
Question 17 (***)
Solve the following logarithmic equation
2 2log (2 1) 2 logz z+ = +.
C2O, 12z=
Created by T. Madas
Created by T. Madas
Question 18 (***)
Solve the following logarithmic equation for y.
2log log (5 24) log 4a a ay y- - =.
8, 12x=
Question 19 (***)
It is given that x satisfies the logarithmic equation log 2(log log 2)a a ax k= -, where0k>, 0a>, 1a≠.
a) Find x in terms of k, giving the answer in a form not involving logarithms.Suppose instead that
x satisfies ()log 5 1 4 log 3x xy+ = + where0x>, 1x≠ and 0y>, 1y≠.
b) Solve the above equation expressing y in terms of x, giving the answer in a form not involving logarithms. 2 4 kx=, 43 15 xy-=
Created by T. Madas
Created by T. Madas
Question 20 (***)
Solve the following logarithmic equation
5log (125 ) 4x=.
5x=Question 21 (***)
Solve the following logarithmic equation
()5 51 2log log 16 3x x+ = -.13,5x x= =
Created by T. Madas
Created by T. Madas
Question 22 (***)
Every £1 invested in a saving scheme gains interest at the rate of 5% per annum so that the total value of this £1 investment after t years is £y.This is modelled by the equation
1.05ty=, 0t≥.
Find after how many years the investment will double.14.2t≈
Question 23 (***)
Solve each of the following logarithmic equations. a) log 16 log 9 2x x= +. b) log 27 3 log 8y y= +.4 4,3 3x x= ≠ -, 32y=
Created by T. Madas
Created by T. Madas
Question 24 (***)
Solve each of the following equations, giving the final answers correct to three significant figures, where appropriate. a) 2 3 900x× =. b) ()()2 2log 7 1 3 log 1y y- = + -.C2P, 5.56x≈, 7y=
Question 25 (***+)
Simplify fully
1 2log 3 log 4n n+ +,
giving the final answer as a single logarithm. log (36 )nnCreated by T. Madas
Created by T. Madas
Question 26 (***+)
Solve each of the following exponential equations, giving the final answers correct to3 significant figures.
a) 2 1 3005 4x-=. b) 11022 y yC2M, 130x≈, 1.16y≈
Question 27 (***+)
Solve the following logarithmic equation
2 22 2log ( 4 3) 4 log ( )w w w w+ + = + +, 1w≠ -.
15w=Created by T. Madas
Created by T. Madas
Question 28 (***+)
Solve the following exponential equation
1 1 6 2 xÄ Ô=Å ÕAE Ö, giving the answer as single logarithm of base 2.2 2log 6 1 log 3x= = +
Question 29 (***+)
Solve the following simultaneous logarithmic equations ()22log 0xy= ()22log 3x y=.C2U, 14,2x y= =
Created by T. Madas
Created by T. Madas
Question 30 (***+)
Solve the following logarithmic equation
3 32log 1 log 7t t= +.
21, 0t t= ≠
Question 31 (***+)
Solve the following logarithmic equation
3 3log 8 3log 3t- =.
C2E, 23t=
Created by T. Madas
Created by T. Madas
Question 32 (***+)
Solve the following logarithmic equation
5 5log (4 ) 2log 1w w- - =.
C2A, 4, 15w w= ≠ -
Question 33 (***+)
Simplify fully the following logarithmic expression, showing clearly all the workings. ()()()log 10 3 10 log 10 90 90 log 10 90 90+ + + + + - +. 1Created by T. Madas
Created by T. Madas
Question 34 (***+)
Solve the following logarithmic equation
2 2 2log log (3 4) 2log (3 4)y y y+ + = -.
24,3y y= ≠
Question 35 (***+)
Solve the following logarithmic equation
2 2log (6 ) 3 logx x- = -.
2, 4x=
Created by T. Madas
Created by T. Madas
Question 36 (***+)
Solve the following logarithmic equation
4 3log log 9x=.
16x=Question 37 (***+)
Solve each of the following equations.
a)1222 3 23.43x+× =.
b) ()()()5 5 5log 2 log 4 3 2log 2 1y y y+ + - = +.0.480x≈, 7y=
Created by T. Madas
Created by T. Madas
Question 38 (***+)
The population P of a certain town in time t years is modelled by the equation10 , 0ktP A t= × ≥,
whereA and k are non zero constants.
When3t=, 19000P= and when 6t=, 38000P=.
Find the value of
A and the value of k, correct to 2 significant figures.C2V, 9500, 0.10A k= =
Question 39 (***+)
Solve the following logarithmic equation
3 32log log ( 2) 2x x- - =.
C2F, 3, 6x=
Created by T. Madas
Created by T. Madas
Question 40 (***+)
Solve the following logarithmic equation
2 12 3log 4 log 27x x-=.
3x= -Question 41 (***+)
Given that 0a≠, 0b≠, 0y≠ and
()22 log 3log 2loga a ab y a y+ + =, express y in terms of a and b, in a form not involving logarithms. 2ayb=Created by T. Madas
Created by T. Madas
Question 42 (***+)
22log 1 log(10 )xx yy
Ä Ô- =Å ÕAE Ö, 0x≠, 0y≠.
Find the exact value of
y. 31100y=
Question 43 (***+) non calculator
The points P and Q lie on the curve with equation
2 26log log 7y x= -, 0x>.
The x coordinates of P and Q are 3 and6, respectively.Find the gradient of the straight line segment
PQ. 2Created by T. Madas
Created by T. Madas
Question 44 (***+)
3 2xy= ×.
a) Describe the geometric transformation which maps the graph of the curve with equation2xy=, onto the graph of the curve with equation 3 2xy= ×.
b) Sketch the graph of 3 2xy= ×.The curve with equation
2xy-= intersects the curve with equation 3 2xy= × at the
point P. c) Determine, correct to 3 decimal places, the x coordinate of P. C2R, vertical stretch by scale factor 3, 0.792x≈ -Created by T. Madas
Created by T. Madas
Question 45 (***+)
It is given that 6log 25p= and 6log 2q=.
Express in terms of
p and qeach of the following expressions a) 6log 200 b) 6log 3.2 c) 6log 75 MP1-F, 6log 200 3p q= +, 61log 3.2 42p q= - +, 6log 75 1p q= - +Created by T. Madas
Created by T. Madas
Question 46 (****)
13xy-=, x??.
a) Sketch the graph of 13xy-= showing the coordinates of all intercepts with the coordinate axes. b) Find to 3 significant figures the x coordinate of the point where the curve13xy-= intersects with the straight line with equation 10y=.
c) Determine to 3 significant figures the x coordinate of the point where the curve13xy-= intersects with the curve 2xy=.
3.10, 2.71
Question 47 (****)
Solve the following logarithmic equation
2 4 1616log 4log 2log 37x x x+ + =, 0x>.
4x=Created by T. Madas
Created by T. Madas
Question 48 (****)
In 1970 the average weekly pay of footballers in a certain club was £100.The average weekly pay,
£P, is modelled by the equation
tP A b= ×, where t is the number of years since 1970, and A and b are positive constants. In1991 the average weekly pay of footballers in the same club had risen to £740.
a) Find the value of A and show that 1.10b=, correct to three significant figures. b) Determine the year when the average weekly pay of footballers in this club will first exceed£10000.
C2Q, 100A=, 2019
Created by T. Madas
Created by T. Madas
Question 49 (****)
Solve each of the following equations, giving the final answers correct to three significant figures, where appropriate. a) 3 26 30x+=. b) ()()4 4log 12 5 log 1 2y y+ - - =. c) 28 8 6 0t t- - =.0.0339x≈ -, 110.39328y= ≈, 0.528t≈
Created by T. Madas
Created by T. Madas
Question 50 (****)
Solve the following simultaneous equations, giving your answers as exact fractions2 2log log 4y x= +
2 38 4y x+=.
324,22 11x y= =
Question 51 (****)
Show clearly that
5 5 25 5log 6 2log 2 log 9 3log 2+ - =.
proofCreated by T. Madas
Created by T. Madas
Question 52 (****)
Solve the following logarithmic equation
10 10 10log ( 4) log ( 16) 1 2logx x x+ + + = +.
164,9x x= ≠ -
Question 53 (****)
Simplify
4 27log 8 log 3-,
giving the final answer as a simplified fraction. 76Created by T. Madas
Created by T. Madas
Question 54 (****)
Solve each of the following equations.
a) ( )4316 1.892
x- b) ()()()2 2 2log 8 1 2log 1 3 log 4y y y- - + = - +.C2D, 9.00x≈, 45y=
Question 55 (****)
Simplify
1221log 8 log8+,
giving the final answer as an integer. 6-Created by T. Madas
Created by T. Madas
Question 56 (****)
Given that log 16ba=, express log (8 )bb in terms of a. 314a+Question 57 (****)
1log3ay= and 8log 1a x= +.
Show clearly that
12xy+=
proofCreated by T. Madas
Created by T. Madas
Question 58 (****)
It is given that
6log 25p= and 6log 2q=.
Simplify each of the following logarithms, giving the final answers in terms of p, q and positive integers, where appropriate. i. ()6log 200. ii. ()6log 3.2. iii. ()6log 75.SYN-B, 3p q+, 142q p-, 1p q+ -
Created by T. Madas
Created by T. Madas
Question 59 (****)
Solve the following exponential equation, giving the answer correct to 3 s.f.22 2 6 0x x- - =.
1.58x≈
Question 60 (****)
Two curves 1C and 2C are defined for all values of x and have respective equations18xy= and 22 3xy= ×.
Show that the
x coordinate of the point of intersection of the two curves is given by 2 13 log 3-.
proofCreated by T. Madas
Created by T. Madas
Question 61 (****)
Solve the following logarithmic equation
2 4log log 2x=.
2x=Created by T. Madas
Created by T. Madas
Question 62 (****)
The functions f and g are defined as
( )()3 2 1xf x-= -, x??, 0x≥ ()2logg x x=, x??, 1x≥. a) Sketch the graph of f. Mark clearly the exact coordinates of any points where the curve meets the coordinate axes. Give the answers, where appropriate, in exact form in terms of logarithms base 2. Mark and label the equation of the asymptote to the curve. b) State the range of f. c) Find ()()f g x in its simplest form.Created by T. Madas
Created by T. Madas
Question 63 (****)
Solve the following logarithmic equation
3 9log log 27x=.
3 3x=Question 64 (****)
The points ()2,10 and ()6,100 lie on the curve with equation ny ax=, where a and n are non zero constants.Find, to three decimal places, the value of
a and the value of n.MP1-H, 2.339a=, 2.096n≈
Created by T. Madas
Created by T. Madas
Question 65 (****)
Solve the following exponential equation, giving the answer correct to 3 s.f. ()4 3 2 10 0y y- - =.C2N, 2.32y≈
Question 66 (****)
Solve the following logarithmic equation
23 9log log 3x x- =.
9, 0x x= ≠
Created by T. Madas
Created by T. Madas
Question 67 (****)
Show that 4x= and 8y= is the only solution pair of the following logarithmic simultaneous equations2 2log (3 4) 1 logx y+ = +.
2 22log 3logy x=.
proofCreated by T. Madas
Created by T. Madas
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