What is x if log2 (3+log3(x))=2? 2,3 are bases of logarithms

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shaznl1's profile pic

shaznl1 | High School Teacher | (Level 1) Salutatorian

Posted on

If log 2 (3+ log 3 (x))= 2

then it means that 2^2=(3+log 3 x)

2^2 =4

4=3+log 3 (x)

Subtract three to both sides

1= log 3 (x)

using the definition of logs again

3^1 = x

x= 3

The only value accepted was a solution to this eqaution is x=3

 

Actually, log just mean the opposite of exponents, so when you take off a log a where a as the base, exponent the other side with "a".

You can checkyour answer by plugging it in and see if it matches the definition of logs

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

First, we'll impose constraints of existence of logarithms:

x>0

3 + log3 (x)>0

log3 (x) > -3

log3 (x) > log3 (3^-3)

log3 (x) > log3 (1/3^3)

x > 1/27

Now, we'll solve the equation taking antilogarithm:

3 + log3 (x) = 2^2

We'll subtract 3 both sides:

log3 (x) = 4 - 3

log3 (x) = 1

We'll take antilogarithm again:

x = 3^1

x = 3

Since x = 3 > 1/27, therefore the value of x is accepted as solution of the given equation: x = 3.

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