Worksheet A EXPONENTIALS AND LOGARITHMS PMT. 1 Express each of the following in the form log a b = c. a 10 3 = 1000 b 3 4 = 81 c 256 = 2 8 d 7 0 = 1

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1 C Worksheet A Epress ech of the following in the form log = c. 0 = = 8 c 56 = 8 d 7 0 = e = f 5 = g 7 9 = 9 h 6 = 6 Epress ech of the following using inde nottion. log 5 5 = log 6 = 4 c 5 = log d log = 0 e = log 9 f lg 0.0 = g log 8 = h log 6 6 = Without using clcultor, find the ect vlue of log 7 49 log 4 64 c log 8 d log 7 e log 5 65 f log 8 8 g log 7 h log 5 5 i log 9 j lg 0.00 k log 6 l log 4 8 m log 9 4 n log o log 5 5 p log Without using clcultor, find the ect vlue of in ech cse. log 5 5 = log = 6 c log 64 = d lg = e log 6 = f log 5 = g log 9 = h lg 0 = i log 7 = j log 4 =.5 k log 0. = l log 8 + = 0 5 Epress in the form log n log 4 + log 7 log 0 log 5 c log 6 d log 9 log e log 5 + log f log 48 log log 9 6 Epress in the form p log q log q 5 log q 5 c log q d log q e 4 log q f log q + log q 5 g log q + loq q h log q log q 4 7 Epress in the form lg n lg 5 + lg 4 lg lg 6 c lg d 4 lg lg 9 e lg 6 lg f + lg g lg h lg 40 8 Without using clcultor, evlute log 54 log log log 5.5 c log 6 + log 7 d log log 6 9 e log log 4 f log 4 8 log 4 9 g log log h lg + lg 5 i log 8 log 8 j log log 5 5 k log 9 5 ( 6 ) + log 5 0 l log 5 log 6 log ( 4 )

2 C Worksheet B Epress in the form p log 0 + q log 0 log 0 log 0 7 c log 0 e log 0 () f log 0 g log 0 d log 0 5 h log 0 Given tht y = log q 8, epress ech of the following in terms of y. log q 64 log q c log q 6 q d log q 4q Given tht = lg nd = lg, epress ech of the following in terms of nd. lg 8 lg 96 c lg 6 9 d lg 6 lg 8 e lg 6 f lg 6 + lg 8 g 4 lg lg 6 h lg 60 + lg 0 4 Without using clcultor, evlute log log 5 4 log 4 + log 8 c log 4 + log 4 log 8 log d 7 7 log 5 log e log 7 log 7 7 f 5 Solve ech eqution, giving your nswers correct to significnt figures. log =.8 log 5 = 0. c log 8 ( ) =. d log 4 ( + ) =. e 5 log y = 9.7 f log 6 ( 5t) + 4. =.6 6 Epress in the form log [f()] 5 log log + log ( + 4) c log + 5 log 5 d log ( ) 4 log e log ( ) log ( + ) f log log 4 + log 7 Solve ech of the following equtions. log + log 5 = log ( + ) log 9 + log 9 0 = c log 4 log 4 ( ) = log 4 + e log 6 = log 6 ( 5) + log 6 5 f log 7 4 = log 7 5 d log 5 5 log 5 ( + ) = log 5 ( + 6) log Solve ech pir of simultneous equtions. log y = log 5 log 5 y = log 5 y = 7 + y = c log = log y d log y = + 5 log y = + y = 0 e log + log = log y f log 0 y + log 0 = + y = 0 log y log =

3 C Worksheet C Find, to significnt figures, the vlue of log 0 60 log 0 6 c log 0 5 d log Solve ech eqution, giving your nswers to deciml plces. 0 = 4 (0 ) 8 = 0 c 0 = 49 d 0 4 = e 0 + = 0 f 00 5 = 0 Show tht log = log log c c, where, nd c re positive constnts. 4 Find, to significnt figures, the vlue of log 7 log 0 7 c log 5 49 d log Solve ech eqution, giving your nswers to significnt figures. = = 0.7 c 8 y = d 4 0. = 0 e 5 t + = 4 f = 0 g = h 5( + ) = 6 i 4 =.7 j 5 = 6 k 7 y + = 9 y + m l 4 5 = = 0 n y = y + 5 o = 7( 4 ) p = Solve the following equtions, giving your nswers to deciml plces where pproprite. + 6 = 0 5( ) + 4 = 0 c 5 + = 8(5 ) d (4 ) + (4 ) = 7 e y + + 7( y ) 5 = 0 f + 7( ) + 0 = 0 g 5 t + 5 t + 4 = 0 h = ( + ) i (6 ) = 0 7 Sketch ech pir of curves on the sme digrm, showing the coordintes of ny points of intersection with the coordinte es. y = y = c y = 4 d y = y = 5 y = ( ) y = 4 y = + 8 A curve hs the eqution y = + where is constnt nd >. Sketch the curve, showing the coordintes of ny points of intersection with the coordinte es nd the equtions of ny symptotes. Given lso tht the curve psses through the point (, 9), find the vlue of. 9 y y = 5 O B A The digrm shows the curve with eqution y = 5 which intersects the coordinte es t the points A nd B. Find the length AB correct to significnt figures.

4 C Worksheet D Given tht = log 0 nd = log 0, find epressions in terms of nd for log 0.5, log 0 4, c log () () Find, to n pproprite degree of ccurcy, the vlues of for which 4 log 5 = 0, log 5 log = 4. () Given tht p = log q, find epressions in terms of p for i log q, ii log 8q. Solve the eqution log 8q log q = log 9. 4 An initil investment of 000 is plced into svings ccount tht offers.% interest every months. The mount of money in the ccount, P, t the end of t yers is given y P = t Find, to the nerest yer, how long it will tke for the investment to doule in vlue. 5 y y = ( ) 4 O y = k The digrm shows the curve with eqution y = ( ) 4. Write down the coordintes of the point where the curve crosses the y-is. () The curve hs n symptote with eqution y = k. Write down the vlue of the constnt k. () c Find the -coordinte of the point where the curve crosses the -is. 6 Solve the eqution log ( + ) log ( ) =. Find, in terms of logrithms to the se 0, the ect vlue of such tht + = 4. 7 Given tht t =, write down epressions in terms of t for i, ii +. Hence solve the eqution + 4( ) + 6 = 0. (5)

5 C Worksheet D continued 8 Find the vlues of for which log ( + 5) + log 5 5 = 7, log ( + ) = 5 log ( ). (5) 9 Given tht log ( + 4) = log 4 + log 5, nd tht log (y + ) = log log (y + ), where y > 0, find the vlue of, the vlue of y, c the vlue of the logrithm of to the se y. () 0 A colony of fst-reeding fish is introduced into lrge, newly-uilt pond. The numer of fish in the pond, n, fter t weeks is modelled y n = c t Find the initil numer of fish in the pond. () Given tht there re 600 fish in the pond fter weeks, use this model to show tht c =, c find the time tken for the initil popultion of fish to doule in size, giving your nswer to the nerest dy. Given tht y = log 8, find epressions in terms of y for i log 8, ii log. Hence, or otherwise, find the vlue of such tht log 8 + log = 6. Solve the simultneous equtions log y = log ( ) + log 4 + log 4 y = (8) Sketch on the sme digrm the curves y = + nd y = ( ), showing the coordintes of ny points where ech curve meets the coordinte es. Given tht the curves y = + nd y = ( ) intersect t the point A, show tht the -coordinte of A is solution of the eqution + = 0, () c hence, show tht the y-coordinte of A is ( 5 + ). 4 Show tht = is solution of the eqution 4( ) = 0. (I) () Show tht using the sustitution u =, eqution (I) cn e written s u 4u + u + 6 = 0. () c Hence find the other rel solution of eqution (I) correct to significnt figures. (7)