Solve x in logarithm equationI am trying to solve x for 2 l o g...

Alvin Parks

Alvin Parks

Answered

2022-11-23

Solve x in logarithm equation
I am trying to solve x for 2 l o g 10 ( x 4 ) l o g 10 4 ( x 1 ) = 0
I have the key with the answer 10 and have confirmed this is correct using Wolfram Alpha but which steps should I take to reach that answer?

Answer & Explanation

dobradisamgn

dobradisamgn

Expert

2022-11-24Added 17 answers

Use the properties of logarithms
k log a x = log a ( x k )
log a x log a y = log a x y
log a x = 0 x = 1
Uriah Molina

Uriah Molina

Expert

2022-11-25Added 7 answers

Start with original problem:
2 log 10 ( x 4 ) log 10 4 ( x 1 ) = 0
A rule of logarithm: 2 log 10 ( x 4 ) = log 10 ( x 4 ) 2
log 10 ( x 4 ) 2 log 10 ( 4 x 4 ) = 0
Another rule of logarithm: Express as the left side of the equation as a single product:
log 10 ( x 4 ) 2 4 x 4 = 0
Convert to exponential form:
10 log 10 ( x 4 ) 2 4 x 4 = 10 0
or
( x 4 ) 2 4 x 4 = 1
and when algebraically rearranging
x 2 8 x + 16 = 4 x 4
or
x 2 12 x + 20 = 0
which is now a quadratic equation that you can solve.

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