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Worked solution

Respect the logarithm base

Solve the equation log⁡x(2x+3)=2\log_x(2x+3)=2.

CJC · Discussion · Question 10

Answer

x=3x=3.

Convert to an exponential equation

A real logarithm requires x>0x>0, x≠1x\ne1 and 2x+3>02x+3>0. The equation gives 2x+3=x22x+3=x^2, so (x−3)(x+1)=0(x-3)(x+1)=0.

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Check the candidates

x=−1x=-1 is an invalid logarithm base. The valid candidate x=3x=3 gives log⁡39=2\log_3 9=2.

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Common mistakes

A logarithm base must be positive and different from 1.

Read the hint