Final answer:
The expression 2log_5(6) + 3log_5(x) + 7log_5(z) simplifies to log_5(36x^3z^7) by applying the power rule and the product rule for logarithms.
Step-by-step explanation:
To use the properties of logarithms to simplify 2log56 + 3log5x + 7log5z, we can apply the power rule for logarithms, the product rule for logarithms, and the quotient rule for logarithms. These properties allow us to manipulate logarithmic expressions. The power rule states that the logarithm of a number raised to an exponent is the exponent multiplied by the logarithm of the number: logb(an) = n · logb(a). Using this, we can simplify the expression as follows:
- Step 1: Apply the power rule
2log56 + 3log5x + 7log5z = log5(62) + log5(x3) + log5(z7) - Step 2: Apply the product rule
log5(62) + log5(x3) + log5(z7) = log5(62 · x3 · z7)
The final expression is log5(36x3z7). This simplifies the original log expression using the properties of exponents and logarithms.