Question

Rewrite the following expression using the properties of logarithms. log2z+2log2x+4log9y+12log9x−2log2y

Answer 1

Use the rules of logarithm:
1. log(x)+log(y)=log(xy)
    log(x)-log(y)=log(x/y)
2. k*log(x) = log(x^k)
    log(x)/k = log(x^(1/k))

log(2z)+2log(2x)+4log(9y)+12log(9x)−2log(2y)
=log(2z)+log(4x^2)+log(9^4y^4)+log(9^12x^12)−log(4y^2)
=log(2z)+log(4x^2)+log(6561y^4)+log(282429536481x^12)−log(4y^2)
=log(59296646043258912 * x^14 * y^6 * z)

Answer 2

The equivalent expression of the logarithmic expression is log(2 *9¹⁶x¹⁴y²z)

How to rewrite the expression?

The logarithmic expression is given as;

log(2z) + 2log(2x) + 4log(9y) + 12log(9x) − 2log(2y)

Express as exponents

log(2z) + log(2x)² + log(9y)⁴ + log(9x)¹² − log(2y)²

Apply the product rule of logarithm

log(2z * (2x)²  * (9y)⁴  * (9x)¹²) − log(2y)²

Evaluate

log(2z * 4x²  * 9⁴y⁴  * 9¹²x¹²) − log(2y)²

Apply the quotient rule of logarithm

log(2z * 4x²  * 9⁴y⁴  * 9¹²x¹²/(2y)²)

Evaluate

log(2z *x²  * 9⁴y²  * 9¹²x¹²)

This gives

log(2 *9¹⁶x¹⁴y²z)

Hence, the equivalent expression is log(2 *9¹⁶x¹⁴y²z)

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