# How do you calculate log_10 (7)?

Nov 16, 2016

You can use Newton's method to find approximations...

#### Explanation:

${\log}_{10} \left(7\right)$ is an irrational number with no simpler representation.

Here's one way to find numerical approximations for it without the benefit of a $\ln$ or $\log$ function...

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Newton's method

Define $f \left(x\right) = {10}^{x} - 7$

Then $f ' \left(x\right) = {10}^{x} \cdot \ln \left(10\right)$

Starting with approximation ${a}_{0} = 1$, use Newton's method, iterating using the formula:

${a}_{i + 1} = {a}_{i} - \frac{f \left({a}_{i}\right)}{f ' \left({a}_{i}\right)} = {a}_{i} - \frac{{10}^{{a}_{i}} - 7}{{10}^{{a}_{i}} \cdot \ln \left(10\right)}$

Of course this requires that you are able to calculate ${10}^{x}$ and know a reasonable approximation for $\ln \left(10\right)$ (say $2.3026$).

For example, if we use $\ln \left(10\right) \approx 2.3026$ then the iterates look like this:

$1.00000000000000$
$0.86971249891427$
$0.84578273695874$
$0.84509858389730$
$0.84509804001812$
$0.84509804001426$
$0.84509804001426$

Note that we do not need to know $\ln \left(10\right)$ very accurately in order to find ${\log}_{10} \left(7\right)$ - it just affects the rate of convergence.