# Math Help - [SOLVED] Basic Logarithms

1. ## [SOLVED] Basic Logarithms

If $log_4(5)=-(3 over (2x))$ , find the value of $log_0.04(8)=$

$\text{If }\log_4(5) \:=\:-\frac{3}{2x}$ .[1]

. . $\text{find the value of: }\:\log_{0.04}(8)$

We have: . $\log_{0.04}(8) \:=\:P$ .[2]

. . $(0.04)^P \:=\:8 \quad\Rightarrow\quad \left(\frac{1}{25}\right)^P \:=\:8 \quad\Rightarrow\quad \left(\frac{1}{5^2}\right)^P \:=\:8$

. . $\left(5^{-2}\right)^P \:=\:8 \quad\Rightarrow\quad 5^{-2P} \:=\:8$

Take logs, base 4: . $\log_4\left(5^{-2P}\right) \:=\:\log_4(8)$

. . $-2P\!\cdot\!\log_4(5) \:=\:\log_4\left(4^{\frac{3}{2}}\right) \quad\Rightarrow\quad -2P\!\cdot\!\log_4(5) \:=\:\frac{3}{2}$

Substitute [1]: . $-2P\left(-\frac{3}{2x}\right) \:=\:\frac{3}{2} \quad\Rightarrow\quad P \:=\:\frac{x}{2}$ .[3]

Equate [2] and [3]: . $\log_{.004}(8) \;=\;\frac{x}{2}$