Question

Use the Laws of logarithms to rewrite the expression

log (x¹⁶ y¹⁷/z²⁰)

in a form with no logarithm of a product, quotient or power. After rewriting we have

log (x¹⁶ y¹⁷/z²⁰) = A log (x) + B log(y) + C log(z)

with
A = ___________

Answer 1

To rewrite the expression log (x¹⁶ y¹⁷/z²⁰), we use the laws of logarithms, resulting in the expression 16 log x + 17 log y - 20 log z. Consequently, the coefficient A in A log (x) + B log(y) + C log(z) is 16.

Step-by-step explanation:

We are asked to rewrite the expression log (x¹⁶ y¹⁷/z²⁰) using the laws of logarithms and identify the value of the coefficient A when expressed in the form A log (x) + B log(y) + C log(z). To do so, we will apply the properties of logarithms that handle logarithms of products, quotients, and powers.

First, we use the property that the logarithm of a product can be expressed as the sum of the logarithms: log xy = log x + log y. Applying this to the xy part of our expression, we get log x¹⁶ + log y¹⁷.

Second, we apply the property that the logarithm of a quotient is the difference of the logarithms: log (a/b) = log a - log b. Thus, the initial expression can be rewritten as log x¹⁶ + log y¹⁷ - log z²⁰.

Lastly, we use the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number: log a^n = n log a. Therefore, our expression becomes 16 log x + 17 log y - 20 log z.

Hence, in the form A log (x) + B log(y) + C log(z), the value of A is 16.