Simplify/Condense

Problem

(log_8)(8*x)+(log_8)(6*x2)−(log_8)(3*x3)

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*x2)−(log_8)(3*x3)

2. Multiply the terms inside the first logarithm.

(log_8)(48*x3)−(log_8)(3*x3)

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

(log_8)((48*x3)/(3*x3))

4. Simplify the fraction inside the logarithm by dividing the coefficients and canceling the x3 terms.

(log_8)(16)

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

(log_2)(2)

6. Apply the change of base formula or the property (log_an)(bm)=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*x2)−(log_8)(3*x3)=4/3

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