Logarithm 2: Identities

Logarithmic identities

Fill in the Blanks

What is the following expression equal to:

  $3^{\log_37}$  ?

Fill in the Blanks

What is the following expression equal to:

  $2^{\log_23}$  ?

Fill in the Blanks

What is the following expression equal to:

  $5^{\log_5\left(3\times2\right)}$  ?

Fill in the Blanks

What is the following expression equal to:

  $5^{\log_53}\times5^{\log_52}$  ?

Fill in the Blanks

What is the following expression equal to:

  $5^{\log_53+\log_52}$  ?

Identity 1

•  $\log_5\left(3\times2\right)=\ \log_5\ 3+\ \log_{5\ }2$  

•  $\log_a\left(xy\right)=\ \log_a\ x+\ \log_{a\ }y$  

• Proof:  raise the base 'a' to these powers and due the monotonity of the expoential function

Fill in the Blanks

What is the following expression equal to:

  $2^{\log_2\left(12\div4\right)}$  ?

Fill in the Blanks

What is the following expression equal to:

  $\frac{2^{\log_2\left(12\right)}}{2^{\log_2\left(4\right)}}$  ?

Fill in the Blanks

What is the following expression equal to:

  $2^{\log_2\left(12\right)\ -\ \log_2\left(4\right)}$  ?

Identity 2

•  $\log_2\left(12\div4\right)=\ \log_2\ 12+\ \log_{2\ }4$  

•  $\log_a\left(\frac{x}{y}\right)=\log_ax-\log_ay$  

• Proof:  raise the base 'a' to these powers and due the monotonity of the expoential function

Fill in the Blanks

What is the following expression equal to:

  $5^{\log_5\left(3^2\right)}$  ?

Fill in the Blanks

What is the following expression equal to:

  $5^{\log_5\left(3\times3\right)}$  ?

Fill in the Blanks

What is the following expression equal to:

  $5^{\log_5\left(3\right)\ +\ \log_5\left(3\right)}$  ?

Fill in the Blanks

What is the following expression equal to:

  $5^{2\ \log_5\left(3\right)}$  ?

Identity 2

•  $\log_5\left(3^2\right)=\ 2\log_5\ 3$  

•  $\log_a\left(x^k\right)=k\log_ax$  

• Proof:  raise the base 'a' to these powers and due the monotonity of the expoential function