Simplify/Condense 3-2 log base 4 of x/2-5

Problem

3−2*(log_4)(x/2)−5

Solution

1. Combine the constant terms 3 and −5 to simplify the expression.

3−5=−2

−2−2*(log_4)(x/2)

2. Factor out the common factor of −2 from both terms.

−2*(1+(log_4)(x/2))

3. Rewrite the constant 1 as a logarithm with base 4 to prepare for condensation.

1=(log_4)(4)

−2*((log_4)(4)+(log_4)(x/2))

4. Apply the product rule for logarithms, which states (log_b)(M)+(log_b)(N)=(log_b)(M*N)

(log_4)(4)+(log_4)(x/2)=(log_4)(4⋅x/2)

(log_4)(2*x)

5. Apply the power rule for logarithms, which states n*(log_b)(M)=(log_b)(Mn) to move the coefficient −2 into the exponent.

−2*(log_4)(2*x)=(log_4)((2*x)(−2))

6. Simplify the expression inside the logarithm by applying the negative exponent.

(2*x)(−2)=1/((2*x)2)

1/(4*x2)

Final Answer

3−2*(log_4)(x/2)−5=(log_4)(1/(4*x2))

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