Exponential functions and logarithms: Logarithmic functions
More logarithmic equations
How do you solve the following equation?
\[2\cdot\log_{2}\left(x\right)=3+\log_{2}\left(6 x-72\right)\]
Drag the steps in the correct order.
- Step 1
- Step 2
- Step 3
- Step 4
- Step 5
- Step 6
- Step 7
- Apply #\purple{n}\cdot \log_{\blue{a}}\left(\green{b}\right)=\log_{\blue{a}}\left(\green{b}^\purple{n}\right) #
- Take logarithms together so you get an equation of the form #\log_\blue{a}\left(\green{b}\right)=\log_\blue{a}\left(\purple{c}\right) #
- Write all terms as a logarithm
- Write the equation #\green{b}=\purple{c}#
- Check if the solutions are valid
- Rewrite the equation #\green{b}=\purple{c}# as an equation you can solve using the quadratic formula
- Solve the equation using the quadratic formula
- #\log_{2}\left( x^2\right)=\log_2\left( 8\right)+\log_{2}\left( 6 x-72\right) #
- Both values of #x# yield positive expressions within the logarithms, so they are both solutions to the equation
- #\log_{2}\left( x^2\right)=\log_{2}\left( 48 x-576\right) #
- #x^2=48 x-576#
- #2\cdot\log_{2}\left( x\right)=\log_2\left( 8\right)+\log_{2}\left( 6 x-72\right) #
- #x^2-48x+576=0#
- #x=24\vee x=24#
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