# How do you solve Log(3x–5)=3 ?

Dec 6, 2015

$x = 331 \frac{2}{3}$

#### Explanation:

Two very important things to remember when working with $\log$ function

1. $\log \left(a\right)$ means ${\log}_{10} a$
(The default base for the $\log$ function is $10$).

2. $\textcolor{b l a c k}{{\log}_{b} a = c}$ means $\textcolor{b l a c k}{{b}^{c} = a}$
(Of the two this is the one you really need to memorize and practice using).

Applying this to the given example:
$\log \left(3 x - 5\right) = 3$

means
$\textcolor{w h i t e}{\text{XXX}} {\log}_{10} \left(3 x - 5\right) = 3$

which in turn means
$\textcolor{w h i t e}{\text{XXX}} {10}^{3} = 3 x - 5$

We can simplify this as
$\textcolor{w h i t e}{\text{XXXXX}} 1000 = 3 x - 5$

$\rightarrow \textcolor{w h i t e}{\text{XXX}} 995 = 3 x$

$\rightarrow \textcolor{w h i t e}{\text{XXX}} x = 331 \frac{2}{3}$

Again, let me emphasize:
the general equivalence
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{{\log}_{b} a = c \iff {b}^{c} = a}$
is something most people do not grasp intuitively and should be memorized and worked with until it comes naturally.