91Ó°ÊÓ

The power rule of logarithms is a very powerful tool. It helps simplify expressions involving logarithms with coefficients. When you have a term like \(a \text{log}_{b}(x)\), you can rewrite it using the power rule: \(\text{log}_{b}(x^a)\). This changes the coefficient into an exponent of the argument inside the logarithm.

Let's break down the example given in the exercise:

• \(3 \text{log}_{7}(x)\) becomes \(\text{log}_{7}(x^3)\)
• \(2 \text{log}_{7}(y)\) becomes \(\text{log}_{7}(y^2)\)
• and \(-5 \text{log}_{7}(z)\) becomes \(\text{log}_{7}(z^{-5})\).

By using the power rule, we turn multiplication outside the logarithm into exponentiation inside the logarithm.

Combining logarithms is another important step based on their properties. When you have the sum of logarithms with the same base, you can combine them into a single logarithm of their product. Similarly, the difference of logarithms can be rewritten as a single logarithm of their quotient. This process requires knowing a few basic logarithm properties:

• The product property: \(\text{log}_{b}(m) + \text{log}_{b}(n) = \text{log}_{b}(mn)\)
• The quotient property: \(\text{log}_{b}(m) - \text{log}_{b}(n) = \text{log}_{b}(m/n)\)

In our exercise, we applied these properties:

• \(\text{log}_{7}(x^3) + \text{log}_{7}(y^2)\) combines into \(\text{log}_{7}(x^3 \times y^2)\)
• \(\text{log}_{7}(x^3 \times y^2) - \text{log}_{7}(z^{-5})\) simplifies into \(\text{log}_{7}((x^3 \times y^2) / z^{-5})\)

These properties simplify the expression by reducing multiple logarithms into a single one.

Finally, let's touch on logarithmic simplification. This involves using multiple properties and rules to simplify our expression fully. In the previous steps, we've reached the expression: \(\text{log}_{7}\frac{x^3 \times y^2}{z^{-5}}\). We can further simplify by considering the exponent rules. Know that dividing by a negative exponent is the same as multiplying by the positive exponent. This helps us turn the expression into a cleaner form: \(\frac{1}{z^{-5}} = z^{5}\).

So, \(\text{log}_{7}\frac{x^3 \times y^2}{z^{-5}} = \text{log}_{7}(x^3 \times y^2 \times z^5)\).

This final step gives us a single, simplified logarithm. It encapsulates the original multiple terms into an easy-to-use format. Simplification helps in solving and understanding more complex logarithmic equations later on.