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Logarithm worksheet

Grades 11–12 · CCSS HSF.BF.B.5, HSF.LE.A.4

Solve each problem and show your work. Round answers to 2 decimal places where needed.

1.
Problem 1.
- Number (x): 1,000
- Base: 2 (binary log)
Find: log₂(1,000)
Answer
2.
Problem 2.
- Number (x): 1,100
- Base: e (natural log)
Find: ln(1,100)
Answer
3.
Problem 3.
- Number (x): 850
- Base: e (natural log)
Find: ln(850)
Answer
4.
Problem 4.
- Number (x): 800
- Base: 10 (common log)
Find: log₁₀(800)
Answer
5.
Problem 5.
- Number (x): 700
- Base: e (natural log)
Find: ln(700)
Answer
6.
Problem 6.
- Number (x): 900
- Base: Custom
- Custom base (b): 5
Find: log₅(900)
Answer
7.
Problem 7.
- Number (x): 1,200
- Base: Custom
- Custom base (b): 4
Find: log₄(1,200)
Answer
8.
Problem 8.
- Number (x): 820
- Base: 2 (binary log)
Find: log₂(820)
Answer
9.
Problem 9.
- Number (x): 1,200
- Base: 2 (binary log)
Find: log₂(1,200)
Answer
10.
Problem 10.
- Number (x): 1,200
- Base: e (natural log)
Find: ln(1,200)
Answer

Answer key: Logarithm worksheet

1. log₂(1,000): 9.97
1. The logarithm base 2 of 1,000 is the exponent y that makes the base raised to y equal 1,000
  $2^{y}$ $= 1,000$
2. Change of base: divide the natural log of x by the natural log of the base
  $log₂ (1,000)$ $= \frac{\ln (1,000)}{\ln (2)}$ $= \frac{6.907755279}{0.6931471806}$ $= 9.9657842847$
3. Check: raising the base to the result gives back x (up to rounding)
  $2^{9.9657842847} \approx 1,000$
2. ln(1,100): 7
1. The logarithm base e of 1,100 is the exponent y that makes the base raised to y equal 1,100
  $e^{y}$ $= 1,100$
2. Change of base: the natural log equals the common (base-10) log of x divided by the common log of e ≈ 2.718282
  $\ln (1,100)$ $= \frac{log₁₀ (1,100)}{log₁₀ (e)}$ $= \frac{3.0413926852}{0.4342944819}$ $= 7.0030654588$
3. Check: raising the base to the result gives back x (up to rounding)
  $e^{7.0030654588} \approx 1,100$
3. ln(850): 6.75
1. The logarithm base e of 850 is the exponent y that makes the base raised to y equal 850
  $e^{y}$ $= 850$
2. Change of base: the natural log equals the common (base-10) log of x divided by the common log of e ≈ 2.718282
  $\ln (850)$ $= \frac{log₁₀ (850)}{log₁₀ (e)}$ $= \frac{2.9294189257}{0.4342944819}$ $= 6.7452363495$
3. Check: raising the base to the result gives back x (up to rounding)
  $e^{6.7452363495} \approx 850$
4. log₁₀(800): 2.9
1. The logarithm base 10 of 800 is the exponent y that makes the base raised to y equal 800
  $10^{y}$ $= 800$
2. Change of base: divide the natural log of x by the natural log of the base
  $log₁₀ (800)$ $= \frac{\ln (800)}{\ln (10)}$ $= \frac{6.6846117277}{2.302585093}$ $= 2.903089987$
3. Check: raising the base to the result gives back x (up to rounding)
  $10^{2.903089987} \approx 800$
5. ln(700): 6.55
1. The logarithm base e of 700 is the exponent y that makes the base raised to y equal 700
  $e^{y}$ $= 700$
2. Change of base: the natural log equals the common (base-10) log of x divided by the common log of e ≈ 2.718282
  $\ln (700)$ $= \frac{log₁₀ (700)}{log₁₀ (e)}$ $= \frac{2.84509804}{0.4342944819}$ $= 6.551080335$
3. Check: raising the base to the result gives back x (up to rounding)
  $e^{6.551080335} \approx 700$
6. log₅(900): 4.23
1. The logarithm base 5 of 900 is the exponent y that makes the base raised to y equal 900
  $5^{y}$ $= 900$
2. Change of base: divide the natural log of x by the natural log of the base
  $log₅ (900)$ $= \frac{\ln (900)}{\ln (5)}$ $= \frac{6.8023947633}{1.6094379124}$ $= 4.2265655051$
3. Check: raising the base to the result gives back x (up to rounding)
  $5^{4.2265655051} \approx 900$
7. log₄(1,200): 5.11
1. The logarithm base 4 of 1,200 is the exponent y that makes the base raised to y equal 1,200
  $4^{y}$ $= 1,200$
2. Change of base: divide the natural log of x by the natural log of the base
  $log₄ (1,200)$ $= \frac{\ln (1,200)}{\ln (4)}$ $= \frac{7.0900768358}{1.3862943611}$ $= 5.1144093452$
3. Check: raising the base to the result gives back x (up to rounding)
  $4^{5.1144093452} \approx 1,200$
8. log₂(820): 9.68
1. The logarithm base 2 of 820 is the exponent y that makes the base raised to y equal 820
  $2^{y}$ $= 820$
2. Change of base: divide the natural log of x by the natural log of the base
  $log₂ (820)$ $= \frac{\ln (820)}{\ln (2)}$ $= \frac{6.7093043403}{0.6931471806}$ $= 9.6794800995$
3. Check: raising the base to the result gives back x (up to rounding)
  $2^{9.6794800995} \approx 820$
9. log₂(1,200): 10.23
1. The logarithm base 2 of 1,200 is the exponent y that makes the base raised to y equal 1,200
  $2^{y}$ $= 1,200$
2. Change of base: divide the natural log of x by the natural log of the base
  $log₂ (1,200)$ $= \frac{\ln (1,200)}{\ln (2)}$ $= \frac{7.0900768358}{0.6931471806}$ $= 10.2288186905$
3. Check: raising the base to the result gives back x (up to rounding)
  $2^{10.2288186905} \approx 1,200$
10. ln(1,200): 7.09
1. The logarithm base e of 1,200 is the exponent y that makes the base raised to y equal 1,200
  $e^{y}$ $= 1,200$
2. Change of base: the natural log equals the common (base-10) log of x divided by the common log of e ≈ 2.718282
  $\ln (1,200)$ $= \frac{log₁₀ (1,200)}{log₁₀ (e)}$ $= \frac{3.079181246}{0.4342944819}$ $= 7.0900768358$
3. Check: raising the base to the result gives back x (up to rounding)
  $e^{7.0900768358} \approx 1,200$