How do you calculate #log_10 (7)#?

1 Answer
Nov 16, 2016

Answer:

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

Explanation:

#log_10(7)# 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...

#color(white)()#
Newton's method

Define #f(x) = 10^x-7#

Then #f'(x) = 10^x*ln(10)#

Starting with approximation #a_0 = 1#, use Newton's method, iterating using the formula:

#a_(i+1) = a_i - (f(a_i))/(f'(a_i)) = a_i - (10^(a_i) - 7)/(10^(a_i)*ln(10))#

Of course this requires that you are able to calculate #10^x# and know a reasonable approximation for #ln(10)# (say #2.3026#).

For example, if we use #ln(10) ~~ 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(10)# very accurately in order to find #log_10(7)# - it just affects the rate of convergence.