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WorkSHEET 5.2 Exponential and
logarithmic functions
1
1
Evaluate log 3   .
3
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1
log 3  
3
 

 log 3 3 1
  log 3  3
 1
2
Simplify log3 (18) + log3 (4)  log3 (2).
log 3 18   log 3  4   log 3  2 
 18  4 
 log 3 

 2 
 log 3  36 
3
Simplify 5 log3 (8)  2 log3 (16) .
5 log 3  8  2 log 3 16 
 log3  8  log3 16 
5
2
 85 
 log3  2 
 16 
 log3 128
4
2
1
Simplify log 4 (27)  log 4 (25) .
3
2
2
1
log 4  27   log 4  25 
3
2
2
1
 log 4  27  3  log 4  25  2
2
1
 log 4  33  3  log 4  52  2
 log 4  9   log 4  5 
 log 4  45 
Maths Quest 11 Mathematical Methods CAS 2nd edition
1
WorkSHEET 5.2 Exponential and logarithmic functions
5
1
Simplify 4log10  x   log10
3
 x   2log  x .
3
10
1
4log10  x   log10
3
 x   2log  x 
3
10
2
 11
 log10  x 4   log10  x 2  3  log10  x3 
 
 1
 log10  x 4   log 10  x 6   log10  x 6 
 
 4 16 
x x
 log10  6 
 x 





 log10  x




6
Solve for x:
log5  2 x  3  4

11
6




11
log10  x 
6
log 5 (2 x  3)  4
2 x  3  54
2 x  3  625
2 x  628
x  314
7
Solve for x:
log x (2)  log x (128)   3
log x  2   log x 128    3
 2  
log x 
 3
 128 

1
x 3
64

4 3x

3
x4
Maths Quest 11 Mathematical Methods CAS 2nd edition
2
WorkSHEET 5.2 Exponential and logarithmic functions
8
Solve 2log3 ( x)  log3 (6 x)  1.
2 log 3  x   log 3  6 x   1
log 3  x 2   log 3  6 x   1
 x2 
log 3    1
 6x 
x2
 31
6x
x 2  18 x
x 2  18 x  0
x  x  18   0
x  0 or x  18
x = 18 is the only solution
(log 3 0 is not possible).
9
32 x  413 x 
Solve for x, correct to 3 decimal places:
32 x  413x 

log10  32 x   log10 4
13 x 

2 x log10  3  1  3 x  log10  4 
2 x log10  3  log10  4   3 x log10  4 
2 x log10  3  3 x log10  4   log10  4 
x  2 log10  3  3log10  4    log10  4 
x
log10  4 
2 log10  3  3log10  4 
x  0.218
Maths Quest 11 Mathematical Methods CAS 2nd edition
3
WorkSHEET 5.2 Exponential and logarithmic functions
10
Sketch the graph of y = 10x, and hence sketch
the graph of y = log10 (x) on the same set of
axes.
The graph of y = 10x has a y-intercept of 1 and
an asymptote at y = 0.
The graph of y = log10 (x) has an x-intercept of
1 and an asymptote at x = 0.
The graphs of y = 10x and y = log10 (x) are
reflections of each other across the line y = x.
Maths Quest 11 Mathematical Methods CAS 2nd edition
4