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Logarithms Practice Test 1

Question : 3 of 40

Marks: +1, -0

\log_{x+3}(x^2+6x+9)+\log_{5x+2}(6x^2-6x)=\log_{2x-1}(8x^3-12x^2+6x-1)

Solution: 👈: Video Solution

Concept:

A logarithm of the form

\log_b(b^n)=n

simplifies directly when the base and argument match a power.

Explanation:

Given:

\log_{x+3}(x^2+6x+9)+\log_{5x+2}(6x^2-6x)=\log_{2x-1}(8x^3-12x^2+6x-1)

Recognize perfect powers:

x^2+6x+9=(x+3)^2

8x^3-12x^2+6x-1=(2x-1)^3

So,

\log_{x+3}(x+3)^2=2

\log_{2x-1}(2x-1)^3=3

Substitute these values:

2+\log_{5x+2}(6x^2-6x)=3

\log_{5x+2}(6x^2-6x)=1

Thus,

6x^2-6x=5x+2

6x^2-11x-2=0

Factor:

(6x+1)(x-2)=0

x=-\frac{1}{6}

x=2

The positive solution is

x=2

Answer:

x=2

(Option A).

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