1. Simplify various logarithmic equations involving logs with different bases and expressions inside and outside of the logarithms. 2. Solve logarithmic equations for the variable inside the logarithm. Common steps include isolating the logarithm and using properties to rewrite the equation in exponential form to solve for the variable. 3. Express logarithmic expressions in terms of given logarithmic values through properties such as logab=logac+logcb.

1. BA101 - exercise
Logarithm
Simplify the logarithmic equations below
1. 2 log a x 3 log a y log a z x2 y3
18. Given log 3 2 0.631 and log 3 5 1.465 . Find:
log a
z
i. log 3 2.5 0.834
2. log 5 125 log 2 32 log 4 64 1
ii. log 3 6 1.631
1
3. log a m 2 log a n log a np3
1
log a
m n 3
19. Find the value of x
3 p3
2
i. 6 x 2 216 5
4. 4 log x log x y 2 log z 2
x z 2
log
y
ii. 2(8 x 1 ) 4(2 x ) 1
x x 1
1 iii. 9 3 81 5
5. log a m log a np 2
1
2 log a n log a
m 3n 3
3 p2 iv. 2x 1 x
(9 ) 27 (9 ) 1
2
6. 4 log x 3 log y log x 2 log z x3 y3
log 2 2x 5
z v. 7 343 4
7. 3 log x log x 2 y 4 log z xz 4
1
log
y
vi. 8x 1 0 3
32 x 8
1 3 vii. log 2 (2 x 1) 3
8. log a x log a y log a z log a
x (2 y ) 3 9
2
2 2 z
2x 5
9. log 5 25 log 4 4 2 log 2 32 viii. 125 5 4
5
ix. log x 3 log x 9 3 3
log 9 3
10. 1
log 2 4 4
x. 73x 5
343 2
3
1 xi. log 3 (5 x 1) 2 2
11. log x b log x a 2c 2 3 log x c log x
a2 b
2 c xii. log 2 3x log 2 (2 x 1) 3 8
13
12. log xyz 2 log y 3 log x x4 z
log x
y xiii. 5 10 1.4306
3x 1 x
1 xiv. 2 5
13. 3 log a x 2 log a xy log a z log a
x5 y 2
1.475
2 z xv. 6 x 2
216 5
3
14. 2 log a a log a ab 3 log a b 1
15. 2 log 2 log 2b 3 log a log
2a 3
b
16. Given log 2 x m and log 2 y n . Express the
following in terms of m and n
i. log 2 x log 2 y m n
ii. log x 2 1
m
17. Given log 3 4 a and log 3 5 b . Express the
following in terms of a and b
i. log 3 80 2a b
ii. log 3 0.75 1 a