Solve for x 1. `log_3(x)^3=(log_3(x))^2` And... 2. `log_2(x^4)=(log_2(x))^2`

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tiburtius | High School Teacher | (Level 2) Educator

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We will use the following property of logarithms (logarithm of a power)

`log_a b^n=n log_a b`                                                        (1)


`log_3 x^3=(log_3 x)^2`

Now by using (1) on the left hand side we get

`3log_3x=(log_3 x)^2`

Now we make substitution `t=log_3x`




From the above line we have 2 solutions `t_1=0` and `t_2=3`. Now we return to our substitution by putting `t_1` and `t_2` instead of `t`.



`x_1=1`  <-- First solution

`3=log_3 x_2`


`x_2=27`  <-- Second solution


This is very similar to previous equation

`log_2 x^4=(log_2x)^2`

Again we use (1) to get


Substitution `t=log_2x`



`t(t-4)=0=> t_1=0,\ t_2=4 `


`2^0=x_1=>x_1=1`  <--First solution


`2^4=x_2=> x_2=16` <--Second solution

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