Laws of logarithms questions with answers

Questions

1. Question 1Evaluate a logarithm

   Without using a calculator, find the value of $\log_{3} 27$.

   Show the worked solution

   1. Step 1: Write the number as a power of 3$27 = 3^{3}$
   2. Step 2: Put it into the log: logₐ(aᵏ) = k$\log_{3} 27 = \log_{3}(3^{3})$$= 3$
2. Question 2Evaluate a logarithm

   Without using a calculator, find the value of $\log_{3} \sqrt{27}$.

   Show the worked solution

   1. Step 1: Write the number as a power of 3 (√ is the power ½)$\sqrt{27} = 3^{\dfrac{3}{2}}$
   2. Step 2: Put it into the log: logₐ(aᵏ) = k$\log_{3} \sqrt{27} = \log_{3}(3^{\dfrac{3}{2}})$$= \dfrac{3}{2}$
3. Question 3Use the laws

   Without using a calculator, find the value of $\lg 500 - \lg 5$.

   Show the worked solution

   1. Step 1: Which law of logarithms combines the two logs into one?Quotient law: $\log_{a} m - \log_{a} n = \log_{a} \dfrac{m}{n}$.
   2. Step 2: Use it to write a single log$\lg 500 - \lg 5 = \lg\left(\dfrac{500}{5}\right)$$= \lg 100$
   3. Step 3: Write the number as a power of 10, then logₐ(aᵏ) = k$= \lg(10^{2})$$= 2$
4. Question 4Use the laws

   Without using a calculator, find the value of $\log_{5} 5 + \log_{5} 25$.

   Show the worked solution

   1. Step 1: Which law of logarithms combines the two logs into one?Product law: $\log_{a} m + \log_{a} n = \log_{a} mn$.
   2. Step 2: Use it to write a single log$\log_{5} 5 + \log_{5} 25 = \log_{5}\left(5 \times 25\right)$$= \log_{5} 125$
   3. Step 3: Write the number as a power of 5, then logₐ(aᵏ) = k$= \log_{5}(5^{3})$$= 3$
5. Question 5Change of base

   Without using a calculator, find the value of $\log_{27} 243$.

   Show the worked solution

   1. Step 1: Change of base to 3: the log of the number ÷ the log of the base$\log_{27} 243 = \dfrac{\log_{3} 243}{\log_{3} 27}$
   2. Step 2: Write both numbers as powers of 3$= \dfrac{\log_{3}(3^{5})}{\log_{3}(3^{3})}$
   3. Step 3: logₐ(aᵏ) = k on the top and the bottom$= \dfrac{5}{3}$
6. Question 6Change of base

   Without using a calculator, find the value of $\log_{25} 5$.

   Show the worked solution

   1. Step 1: Change of base to 5: the log of the number ÷ the log of the base$\log_{25} 5 = \dfrac{\log_{5} 5}{\log_{5} 25}$
   2. Step 2: Write both numbers as powers of 5$= \dfrac{\log_{5}(5^{1})}{\log_{5}(5^{2})}$
   3. Step 3: logₐ(aᵏ) = k on the top and the bottom$= \dfrac{1}{2}$