# How do you calculate log_4 28?

Aug 24, 2016

${\log}_{4} 28 = 2.4037$

#### Explanation:

Using the identity ${\log}_{b} a = \frac{{\log}_{n} a}{{\log}_{n} b}$, if we choose $n = 10$, then

${\log}_{4} 28 = \log \frac{28}{\log} 4$

Now using tables as $\log 28 = 1.44716$ and $\log 4 = 0.60206$

${\log}_{4} 28 = \frac{1.44716}{0.60206} = 2.4037$

Aug 24, 2016

$2.404$

#### Explanation:

Log form and index form are interchangeable, and sometimes one form is easier to use than the other.

Note that: ${\log}_{a} b = c \text{ "harr" } {a}^{c} = b$

"The base stays the base and the other two change around"

${\log}_{4} 28 = x \text{ "harr " } {4}^{x} = 28$

We can see that this is not an integer answer because:

${4}^{2} = 16 \mathmr{and} {4}^{3} = 64$ x is between 2 and 3.

${4}^{x} = 28 \text{ log each side}$

$x \log 4 = \log 28 \text{ power law as well}$

$x = \frac{\log 28}{\log 4} \text{ isolate x}$

$x = 2.404 \text{ calculator or tables}$

The same result would have been obtained using
the change of base law.