Question

Rewrite in terms of simpler logs (expand):

Answer 1

`\textsf{1)} \quad 3\log_a\left(x\right)+8\log_a\left(z\right)-(5)/(6)\log_a\left(y\right)`

`\textsf{2)} \quad \log_a\left((1)/(x^(14)y^(28))\right)`

Explanation:

Question 1

Given logarithmic expression:

`\log_a\left(\sqrt[6]{(x^(18)z^(48))/(y^5)}\right)`

To rewrite the given logarithmic expression in terms of simpler logs, begin by applying the fractional exponent rule:

`\log_a\left(\left((x^(18)z^(48))/(y^5)\right)^{(1)/(6)\right)`

Now, apply the power rule of logarithms:

`(1)/(6)\log_a\left((x^(18)z^(48))/(y^5)\right)`

Apply the quotient rule of logarithms:

`(1)/(6)\log_a\left(x^(18)z^(48)\right)-(1)/(6)\log_a\left(y^5\right)`

Apply the product rule of logarithms:

`(1)/(6)\log_a\left(x^(18)\right)+(1)/(6)\log_a\left(z^(48)\right)-(1)/(6)\log_a\left(y^5\right)`

Apply the power rule of logarithms:

`(18)/(6)\log_a\left(x\right)+(48)/(6)\log_a\left(z\right)-(5)/(6)\log_a\left(y\right)`

Simplify the fractions:

`3\log_a\left(x\right)+8\log_a\left(z\right)-(5)/(6)\log_a\left(y\right)`

`\hrulefill`

Question 2

Given logarithmic expression:

`(3)/(8)\left[16\log_a(x)+32\log_a(y)\right]-5\left[4\log_a(x)+8\log_a(y)\right]`

Apply the power rule of logarithms inside the brackets:

`(3)/(8)\left[\log_a(x^(16))+\log_a(y^(32))\right]-5\left[\log_a(x^(4))+\log_a(y^(8))\right]`

Apply the product rule of logarithms inside the brackets:

`(3)/(8)\log_a(x^(16)y^(32))-5\log_a(x^(4)y^(8))`

Apply the power rule of logarithms:

`\log_a\left((x^(16)y^(32))^{(3)/(8)\right)-\log_a\left((x^(4)y^(8))^5\right)`

Apply the power of a power rule of exponents:

`\log_a(x^(6)y^(12))-\log_a(x^(20)y^(40))`

Apply the quotient rule of logarithms:

`\log_a\left((x^(6)y^(12))/(x^(20)y^(40))\right)`

Simplify the argument:

`\log_a\left((1)/(x^(14)y^(28))\right)`

`\hrulefill`

`\boxed{\begin{array}{rl}\underline{\sf Laws\;of\;Exponents}\\\\\sf Product:&\log_axy=\log_ax + \log_ay\\\\\sf Quotient:&\log_a \left((x)/(y)\right)=\log_ax - \log_ay\\\\\sf Power:&\log_ax^n=n\log_ax\\\\\end{array}}`