Simplify/Condense

Problem

(log_8)(8*x)+(log_8)(6*x^2)−(log_8)(3*x^3)

Solution

1. Apply the product rule for logarithms, which states that (log_b)(M)+(log_b)(N)=(log_b)(M*N) to the first two terms.

(log_8)(8*x⋅6*x^2)−(log_8)(3*x^3)

2. Multiply the terms inside the first logarithm.

(log_8)(48*x^3)−(log_8)(3*x^3)

3. Apply the quotient rule for logarithms, which states that (log_b)(M)−(log_b)(N)=(log_b)(M/N)

(log_8)((48*x^3)/(3*x^3))

4. Simplify the fraction inside the logarithm by dividing the coefficients and canceling the x^3 terms.

(log_8)(16)

5. Evaluate the logarithm by expressing the base and the argument as powers of 2.

(log_2^3)(2^4)

6. Apply the change of base formula or the property (log_a^n)(b^m)=m/n*(log_a)(b)

4/3*(log_2)(2)

7. Simplify using the identity (log_b)(b)=1

4/3

Final Answer

(log_8)(8*x)+(log_8)(6*x^2)−(log_8)(3*x^3)=4/3

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