What is the answer to |log_{5}(x^2)-4|=2+|log_{5}(x)-3| ?

The answer to |log_{5}(x^2)-4|=2+|log_{5}(x)-3| is x= 1/5 ,x=125

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|log_{5}(x^2)-4|=2+|log_{5}(x)-3|

lo g_{5} (x^{2}) - 4 = 2 + ∣ lo g_{5} (x) - 3 ∣

x = \frac{1}{5}, x = 125

Decimal Notation

x = 0.2, x = 125

Solution steps

lo g_{5} (x^{2}) - 4 = 2 + ∣ lo g_{5} (x) - 3 ∣

Apply log rules

∣ 2 lo g_{5} (x) - 4 ∣ = 2 + ∣ lo g_{5} (x) - 3 ∣

Rewrite the equation with

lo g_{5} (x) = u

∣ 2 u - 4 ∣ = 2 + ∣ u - 3 ∣

Solve

∣ 2 u - 4 ∣ = 2 + ∣ u - 3 ∣ : u = - 1 or u = 3

u = - 1, u = 3

Substitute back

u = lo g_{5} (x),

solve for

x

Solve

lo g_{5} (x) = - 1 : x = \frac{1}{5}

Solve

lo g_{5} (x) = 3 : x = 125

x = \frac{1}{5}, x = 125

Verify Solutions:

x = \frac{1}{5}

True

, x = 125

True

The solutions are

x = \frac{1}{5}, x = 125

lo g_{2} (x^{2}) = 8

lo g_{2} (x^{2}) = 2

lo g_{4} (2 x + 1) = 0

lo g_{10} (y) = \frac{1}{2}

lo g_{10} (x + 6) - lo g_{10} (x - 3) = lo g_{10} (4)

What is the answer to |log_{5}(x^2)-4|=2+|log_{5}(x)-3| ?
The answer to |log_{5}(x^2)-4|=2+|log_{5}(x)-3| is x= 1/5 ,x=125
Solve for x: |log_{5}(x^2)-4|=2+|log_{5}(x)-3|
The solution is x= 1/5 ,x=125