Question

Need help for math. Write each expression as a sum or difference or multiples of logarithms assume they’re all positive

Answer 1

We want to rewrite the following logarithm

$\log_b\sqrt{6x}$

To rewrite this logarithm, first let's rewrite the argument of the logarithm. Using the following property

$\sqrt{a}=a^{\frac{1}{2}}$

We can rewrite our expression as

$\operatorname{\log}_b\sqrt{6x}=\operatorname{\log}_b6x^{\frac{1}{2}}$

Using the following property

$\log_nx^m=m\log_nx$

We can rewrite our expression as

$\operatorname{\log}_b6x^{\frac{1}{2}}=\frac{1}{2}\operatorname{\log}_b6x$

and finally, using the logarithm property of a product

$\log_n(x\cdot y)=\log_nx+\log_ny$

We can rewrite our expression as

$\begin{gathered} \frac{1}{2}\operatorname{\log}_b6x=\frac{1}{2}\operatorname{\log}_b(6\cdot x) \\ =\frac{1}{2}(\log_b6+\log_bx) \end{gathered}$

and this is the final form of our expression.

$\operatorname{\log}_b\sqrt{6x}=\frac{1}{2}(\log_b6+\log_bx)$