# How do you simplify log10(sqrt10)?

the equivalent of the expression is $\frac{3}{2}$

#### Explanation:

We start from the given

$\log 10 \sqrt{10}$

which also means

${\log}_{10} 10 \cdot \sqrt{10}$

${\log}_{10} {10}^{1} \cdot {10}^{\frac{1}{2}}$

${\log}_{10} {10}^{\left(1 + \frac{1}{2}\right)}$

${\log}_{10} {10}^{\frac{3}{2}}$

from the Laws of logarithms, we have
${\log}_{b} {b}^{x} = x$

so that the final simplification is simply

$\frac{3}{2}$

Feb 19, 2016

$\log 10$ means ${\log}_{10} \left(10\right)$

#### Explanation:

${\log}_{10} \left(10\right) \left(\sqrt{10}\right)$

${\log}_{a} a = 1$ because if you take the change of base rule ${\log}_{a} n = \left(\log \frac{n}{\log} a\right)$, with a and n being equal, the expressions cancel each other out, giving us 1.

=$1 \times \sqrt{10}$

= $\sqrt{10} \mathmr{and} 2.16$

Practice exercises:

1. Simplify $\frac{{\log}_{3} \frac{3}{\log} _ 8 \left(8\right)}{\sqrt{5}}$. Don't forget to rationalize the denominator.

2. Simplify ${\log}_{x - 1} \left(x + 1\right) + {\log}_{x + 1} \left(x - 1\right)$. Note: a helpful rule is: ${\log}_{a} n + {\log}_{a} m = {\log}_{a} \left(n \times m\right)$