Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Functions - Logarithmic functions - Log laws.
Test Yourself 1.


 

Use the six laws of logarithms as outlined in the video to simplify/solve the following:

Log laws with numbers. 1. Simplify log5 2 + log5 10.
Answer.2log52 + 1.		 2. Simplify log3 9 + log7 49.
Answer.4.
  3. Simplify log4 20 - log4 5.
Answer.1.		 4. Simplify 2 log2 8 - 3log6 216.
Answer.-3.
  5. Simplify log5 50 + log5 10 - log5 4.
Answer.3.		 6. Simplify log4 20 + (log4 32 - log4 10).
Answer.2.
  7. Simplify 5log8 2 + 0.5log8 4.
Answer.2.		 8. Simplify

log10 125 - log10 4 + log10 32.

Answer.3.
Log laws with pronumerals. 9. Simplify log (x + 1) + log (3 - x)
Answer.log(x + 1)(3 - x).		 10. Simplify log (8x) + log (2x)
  11. Simplify log5 8x - log5 2x
Answer.2log52.		 12. Simplify log10 x2y3 - log10 xy
Answer.log (xy2).
  13. Evaluate log4 32 - log4 5 to 3 significant places.
Answer:1.34		 14. Find the value of log5 200 - 3log5 2.
Answer. 2
  15. Simplify 2 log x + 3 log y - log xy2
Answer.log(xy).		 16. Use two approaches to simplify
log5 25.
Answer.2
  17. Evaluate e(ln2 + ln3)
Answer.6.		 18. Evaluate e(2 ln4 - 3 ln2)
Answer.e8
  19. If x = loga 3 and y = loga 5, prove that

.

 20. Simplify
loge (e2 + e) - loge (e+1).
Answer.1
  21. If m = en, show that loge (m2) = 2n. 22. Express 3log2 8 in its simplest form..
  23. If e4x = 4, show that . 24. Simplify log10 20A - log10 2A
Answer.1
  25. If a = log12 b and b > 1, which of the following is equivalent to ?

(a) -log12 b.

(b) logb 12.

(c) .

(d) .


Answer.b		 26. Which expression is equivalent to
4 + log2 x?

(a) log2 (2x).

(b) log2 (16 + x).

(c) 4log2 (2x).

(d) log2 (16x).


Answer.d.
  27. Given that

2log3 (x2y) = 3 + log3 x - log3 y,

express y in terms of x.

Hint.Either make the coefficient for the first term into an index and then collect the log terms in x and y OR
split the LHS into two terms, bring down the index then collect like terms and divide.
Answer.y = 3/x.		 28.
Substitutions. 29. Given loga3 = 1.6 and loga7 = 2.4, find the value of loga (21a).
Answer.5.		 30. Given that log2 5 = 2.32 and that
log2 3 = 1.58, find the value of:

(i) log2 0.6;

(ii) log2 60.

Answer.(i) -0.74.
(ii) 5.90.
  31. Given that loga 2 = 0.4307 and

loga 3 = 0.6826, find the value of loga 24.


Answer.1.9747.		 32. Given that logm p = 1.175 and
logm q = 2.25, find the exact value of

(i) logm (pq);

(ii) ;

(iii) pq2 in terms of m.


Answer.(i) 4.
(ii) -0.5.
(iii) value is m6.25.
  33. Given that log3 x = a and
log3 y = b, express in terms of a and b.
Answer.2 + a/2 - b.		 34. If loga (xy3) = 1 and loga (x2y) = 1,
what is the value of loga (xy)?
Answer.loga(xy) = 0.6 = 3/5.
Equations. 35. Solve 2 loge x = loge (3x + 10).
Answer.x = 5.		 36. Solve for x:

log5 3 = 2log5 6 + log5 x


Answer.x = 5.
  37. Solve log10(x2) + 3 = log10(x5).
Hint.Either remove the indices and collect the terms in log x
OR
The constant (3) is out of place. So convert it to a log - check your laws to see what you can write instead of 1 - which you can then multiply by 3.

Answer.x = 10.		 38. Solve
log2 (x - 1) = 5 - log2 (x + 3).
Hint.Either bring the log term from the right side to the left and combine the log terms
OR
Change the 5 to be a log to the base 2.
Answer.x = 5.
  39. Solve log3 (2x - 1) + log3 (x - 4) = 2.
Answer.x = 5.		 40. Solve ln (x + 12) = 2 ln x
Answer.x = 4.
  41. Solve the pair of simultaneous equations
Answer.x = 1000 and y = 10. 		42. Solve the equation:

2log2 x - log2 (x + 4) = 1

Answer.x = 4 (x ≠ -2).