Simplify/Condense

Problem

(log_5)(x)/3+(log_5)(y)/3+(log_5)(z)/3⋅(log_5)(w)/3

Solution

1. Factor out the common constant 1/3 from the first two terms to group the logarithmic expressions.

1/3*((log_5)(x)+(log_5)(y))+((log_5)(z)*(log_5)(w))/9

2. Apply the product rule for logarithms, (log_b)(M)+(log_b)(N)=(log_b)(M*N) to the terms inside the parentheses.

1/3*(log_5)(x*y)+((log_5)(z)*(log_5)(w))/9

3. Apply the power rule for logarithms, n*(log_b)(M)=(log_b)(Mn) to move the coefficient 1/3 to the exponent.

(log_5)((x*y)(1/3))+((log_5)(z)*(log_5)(w))/9

4. Rewrite the exponent as a cube root for the final condensed form.

(log_5)(√(3,x*y))+((log_5)(z)*(log_5)(w))/9

Final Answer

(log_5)(x)/3+(log_5)(y)/3+(log_5)(z)/3⋅(log_5)(w)/3=(log_5)(√(3,x*y))+((log_5)(z)*(log_5)(w))/9

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